Research ArticleEuler Type Integrals and Integrals in Terms ofExtended Beta Function

Subuhi Khan1 and Mustafa Walid Al-Saad2

1 Department of Mathematics Aligarh Muslim University Aligarh 202002 India2 General Requirements Unit Fatima College of Health Sciences Al Ain UAE

Correspondence should be addressed to Subuhi Khan subuhi2006gmailcom

Received 15 February 2014 Accepted 12 May 2014 Published 12 June 2014

Academic Editor Alfredo Peris

Copyright copy 2014 S Khan and M W Al-Saad This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We derive the evaluations of certain integrals of Euler type involving generalized hypergeometric series Further we establish atheorem on extended beta function which provides evaluation of certain integrals in terms of extended beta function and certainspecial polynomials The possibility of extending some of the derived results to multivariable case is also investigated

1 Introduction

Euler generalized the factorial function from the domain ofnatural numbers to the gamma function

Γ (120572) = int

infin

0

119905120572minus1

119890minus119905

119889119905 (Re (120572) gt 0) (1)

defined over the right half of the complex plane This ledLegendre (in 1811) to decompose the gamma function into theincomplete gamma functions 120574(120572 119909) and Γ(120572 119909) [1]

120574 (120572 119909) = int

119909

0

119905120572minus1

119890minus119905

119889119905 (Re (120572) gt 0) (2a)

Γ (120572 119909) = int

infin

119909

119905120572minus1

119890minus119905

119889119905 (2b)

which are obtained from (1) by replacing the upper andlower limits by 119909 respectively The closed-form solutions toa considerable number of problems in applied mathematicsastrophysics nuclear physics statistics and engineering canbe expressed in terms of incomplete gamma functionsThese functions develop singularities at the negative integersChaudhry and Zubair [2] extended the domain of these func-tions to the entire complex plane by inserting a regularizationfactor exp(minus119887119905) in the integrand of (1) For Re(119887) gt 0 thisfactor clearly removes the singularity coming from the limit119905 = 0 For 119887 = 0 this factor becomes unity and thus we get

the original gamma function We note the following relation[3 page 20 (12)]

Γ119887(120572) = int

infin

0

119905120572minus1 exp(minus119905 minus 119887

119905) 119889119905

= 2(119887)1205722

119870120572(2radic119887) (Re (120572) gt 0)

(3)

where 119870119899(119909) is the modified Bessel function of the second

kind of order 119899 (or Macdonaldrsquos function) [1] The rela-tionships between the generalized gamma and Macdonaldfunctions could not have been apparent in the originalgamma functionThese generalized gamma functions provedvery useful in diverse engineering and physical problems seefor example [2 3] and references therein

We note that Riemannrsquos zeta function 120577(119909) defined by theseries [1 page 85 (298)]

120577 (119909) =

infin

sum

119899=1

1

119899119909(119909 gt 1) (4)

is useful in proving convergence or divergence of otherseries by means of comparison test Zeta function is closelyrelated to the logarithm of the gamma function and tothe polygamma functions The regularizer exp(minus119887119905) alsoproved very useful in extending the domain of Riemannrsquos zetafunction thereby providing relationships that could not havebeen obtained with the original zeta function

Hindawi Publishing CorporationJournal of MathematicsVolume 2014 Article ID 686324 12 pageshttpdxdoiorg1011552014686324

2 Journal of Mathematics

In viewof the fact that the regularization factor is useful inextending the domain of the gamma and zeta functions Thedomain of other special functions could be usefully extendedin a similar manner

In particular Eulerrsquos beta function 119861(120572 120573) has a closerelationship to gamma function that is

119861 (120572 120573) = 119861 (120573 120572) =Γ (120572) Γ (120573)

Γ (120572 + 120573) (5)

and can be usefully extendable We first recall the basic Eulerintegral which defines the beta function as follows

119861 (120572 120573) = int

1

0

119906120572minus1

(1 minus 119906)120573minus1

119889119906 (Re (120572) Re (120573) gt 0)

(6)

Keeping in view the fact that for the 120572-120573 symmetry to bepreserved there must be symmetry of the integrand in 119906 and1 minus 119906 Chaudhry et al [3] introduced an extension of Eulerrsquosbeta function 119861(120572 120573) in the following form

119861 (120572 120573 119887) = int

1

0

119905120572minus1

(1 minus 119905)120573minus1 exp(minus 119887

119905 (1 minus 119905)) 119889119905

(Re (119887) gt 0)

(7)

which for 119887 = 0 reduces to the original beta functionThe transformations 119905 = (119906 + 1)2 and 119905 = (119906 minus 119886)

(119888 minus 119886) in (7) yield the following integral representations forthe extended beta function (EBF) 119861(120572 120573 119887) [3 page 22 (29)(211)]

119861 (120572 120573 119887)

= 21minus120572minus120573

times int

1

minus1

(1 + 119906)120572minus1

(1 minus 119906)120573minus1 exp(minus 4119887

(1 minus 1199062)) 119889119906

(Re (119887) gt 0)

(8)

119861 (120572 120573 119887)

= (119888 minus 119886)1minus120572minus120573

times int

119888

119886

(119906 minus 119886)120572minus1

(119888 minus 119906)120573minus1 exp(minus 119887(119888 minus 119886)

2

(119906 minus 119886) (119888 minus 119906)) 119889119906

(Re (119887) gt 0)

(9)

respectivelyThe EBF 119861(120572 120573 119887) is extremely useful in the sense that

most of the properties of the beta function carry overnaturally and simply for it This extension is also importantdue to the fact that this function is related to other specialfunctions for particular values of the variables

We note the following connections [3]

119861 (120572 minus120572 119887) = 2119890minus2119887

119870120572(2119887) (Re (119887) gt 0) (10)

119861 (120572 120572 119887)

= radic1205872minus120572

119887(120572minus1)2

119890minus2119887

119882minus12057221205722

(4119887)

= radic12058721minus2120572

119890minus4119887

Ψ(1

2 1 minus 120572 4119887) (Re (119887) gt 0)

(11)

where 119882119898119899

(119911) denotes the Whittaker function [1] andΨ(119886 119888 119911) denotes the second form of solutions of Kummerrsquosequation [4]

Also we note that

119861 (0 0 119887) = 2119890minus2119887

1198700(2119887) (Re (119887) gt 0)

119861 (1

2 minus1

2 119887) = radic

120587

119887119890minus4119887

(Re (119887) gt 0)

119861 (1

21

2 119887) = 120587Erfc (2radic119887) (Re (119887) gt 0)

(12)

where Erfc(119911) = (radic1205872)erfc(119911) denotes the complementaryerror function [4]

Explicit evaluations of some integrals of Euler type

int

1

0

119906120572minus1

(1 minus 119906)120573minus1

119891 (119906) 119889119906 (13)

for some particular functions 119891 specially in the symmetriccase 120572 = 120573 are derived in [5] These evaluations are relatedto various reduction formulae for hypergeometric functionsrepresented by such integrals These formulae generalize theevaluations of some symmetric Euler integrals implied by thefollowing result due to Pitman [6]

If a standard Brownian bridge is sampled at time 0time 1 and at 119899 independent random times with uniformdistribution on [0 1] then the broken line approximationto the bridge obtained from these 119899 + 2 values has a totalvariation whose mean square is 119899(119899 + 1)(2119899 + 1)

Motivated and inspired by the work of Ismail and Pitman[5] and Chaudhry et al [3] in this paper we derive theevaluations of certain Euler type integrals and some integralsin terms of EBF 119861(120572 120573 119887) In Section 2 we obtain theevaluations of certain integrals of the following type

119868119889120572120573120574119886119890

[120601 (119905) 120595 (119905)]

=1

119861 (120572 120573)int

119890

119886

(119905 minus 119886)120572minus1

(119890 minus 119905)120573minus1

(120601 (119905))120574

times119903119865119905

[

[

(119891119903)

119889120595 (119905)

(119892119905)

]

]

119889119905

(14)

where119903119865119905denotes the generalized hypergeometric series [4]

for particular functions 120601(119905) and 120595(119905) In Section 3 weestablish a theorem on EBF 119861(120572 120573 119887) and apply it to obtainevaluations of certain integrals in terms of EBF 119861(120572 120573 119887)Finally we give some concluding remarks in Section 4

Journal of Mathematics 3

2 Euler Type Integrals

We derive the evaluations of certain integrals of the followingtype

119868119889120572120573120574119886119890

[120601 (119905) 120595 (119905)]

=1

119861 (120572 120573)int

119890

119886

(119905 minus 119886)120572minus1

(119890 minus 119905)120573minus1

(120601 (119905))120574

times119903119865119905

[

[

(119891119903)

119889120595 (119905)

(119892119905)

]

]

119889119905

(15)

We evaluate the integrals of type (15) for particularfunctions 120601(119905) and 120595(119905) by considering the following cases

Case 1 Taking120601(119905) = (1minus1199091119905)minus1205721(1minus119909

2119905)minus1205722 120595(119905) = 1119905(1minus119905)

119886 = 0 and 119890 = 120574 = 1 in (15) and using [5 page 962 (7)]

int

1

0

119905120572minus1

(1 minus 119905)120573minus1

(1 minus 1199091119905)minus1205721

(1 minus 1199092119905)minus1205722

119889119905

= 119861 (120572 120573) 1198651[120572 120572

1 120572

2 120572 + 120573 119909

1 119909

2]

(Re (120572) Re (120573) gt 0 max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 lt 1)

(16)

where 1198651denotes the Appell function [4] in the rhs

after expanding119903119865119905[

(119891119903)

119889120595(119905)

(119892119905)

] and simplifying we find the

following integral

119868119889120572120573101

[(1 minus 1199091119905)minus1205721

(1 minus 1199092119905)minus1205722

1

119905 (1 minus 119905)]

=

infin

sum

11989811198982=0

(120572)1198981+1198982

(1205721)1198981

(1205722)1198982

1199091198981

11199091198982

2

(120572 + 120573)1198981+1198982

1198981119898

2

times119903+2119865119905+2

[[

[

(119891119903) 1

2(1 minus 120572 minus 120573 minus 119898

1minus 119898

2) 1 minus

1

2(120572 + 120573 + 119898

1+ 119898

2)

4119889

(119892119905) 1 minus (120572 + 119898

1+ 119898

2) 1 minus 120573

]]

]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 lt 1)

(17)

Case 2 Taking 120601(119905) =119901

119865119902[

(119886119901)

119888119905(119890minus119905)

(119887119902)

] 120595(119905) = 1119905(119890 minus 119905)

119886 = 0 and 120574 = 1 in (15) and using [7 page 303 (1)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1

119901119865119902

[[

[

(119886119901)

119888119905 (119890 minus 119905)

(119887119902)

]]

]

119889119905

= (119890)120572+120573minus1

119861 (120572 120573)

times119901+2

119865119902+2

[[[[[

[

(119886119901) 120572 120573

1198881198902

4

(119887119902)

120572 + 120573

2120572 + 120573 + 1

2

]]]]]

]

(119890Re (120572) Re (120573) gt 010038161003816100381610038161003816arg (4 minus 1198881198902)10038161003816100381610038161003816 lt 120587)

(18)

we find the following integral

11986811988912057212057310119890

[

[

119901119865119902

[

[

(119886119901)

119888119905 (119890 minus 119905)

(119887119902)

]

]

1

119905 (119890 minus 119905)

]

]

= (119890)120572+120573minus1

infin

sum

119898=0

(119886119901)119898

(120572)119898(120573)

119898(119888119890

2

4)

119898

(119887119902)119898

((120572 + 120573)2)119898((120572 + 120573 + 1)2)

119898119898

times119903+2119865119905+2

[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 1 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905) 1 minus 120572 minus 119898 1 minus 120573 minus 119898

]]]]

]

(119890Re (120572) Re (120573) gt 0 10038161003816100381610038161003816arg (4 minus 1198881198902)1003816100381610038161003816

1003816lt 120587)

(19)

Case 3 Taking120601(119905) =119901

119865119902[

(119886119901)

119888119905(119890minus119905)

(119887119902)

]120595(119905) = 1(1199052

(119890 minus 119905)2

)

119886 = 0 and 120574 = 1 and proceeding as in Case 2 we find thefollowing integral

11986811988912057212057310119890

[[

[

119901119865119902

[[

[

(119886119901)

119888119905 (119890 minus 119905)

(119887119902)

]]

]

1

1199052(119890 minus 119905)2

]]

]

= (119890)120572+120573minus1

infin

sum

119898=0

(119886119901)119898

(120572)119898(120573)

119898(119888119890

2

4)119898

(119887119902)119898

((120572 + 120573) 2)119898((120572 + 120573 + 1) 2)

119898119898

4 Journal of Mathematics

times119903+6119865119905+8

[[[[[[

[

(119891119903) 119895 minus 120572 minus 120573

21

2(1 minus 120572 minus 120573) minus 119898 1 minus

1

2(120572 + 120573) minus 119898

4119889

1198904

(119892119905) 119895 minus 120572 minus 120573

41 minus 120572 minus 119898

22 minus 120572 minus 119898

21 minus 120573 minus 119898

22 minus 120573 minus 119898

2

]]]]]]

]

(119895 = 1 2 3 4) (119890Re (120572) Re (120573) gt 010038161003816100381610038161003816arg (4 minus 1198881198902)10038161003816100381610038161003816 lt 120587)

(20)

Case 4 Taking 120601(119905) = K(119888radic119905(119890 minus 119905)) 120595(119905) = 1119905(119890 minus 119905) 119886 = 0and 120574 = 1 in (15) where K(119911) is the complete elliptic integralof the first kind [4 page 35 (32)] defined by

K (119911) = int

1205872

0

119889120579

radic(1 minus 1199112sin2120579)(21)

and using [7 page 265 (4)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1K (119888radic119905 (119890 minus 119905)) 119889119905

=120587

2(119890)

120572+120573minus1

119861 (120572 120573)41198653

[[[[[[

[

1

21

2 120572 120573

1198882

1198902

4

1120572 + 120573

2120572 + 120573 + 1

2

]]]]]]

]

(Re (120572) Re (120573) gt 010038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(22)

we find the following integral

11986811988912057212057310119890

[K(119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)

]

=

120587

2

(119890)120572+120573minus1

infin

sum

119898=0

(12)119898(12)

119898(120572)

119898(120573)

119898(1198882

1198902

4)

119898

(1)119898((120572 + 120573)2)

119898((120572 + 120573 + 1)2)

119898119898

times119903+2119865119905+2

[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 1 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905) 1 minus 120572 minus 119898 1 minus 120573 minus 119898

]]]]

]

(Re (120572) Re (120573) gt 0 10038161003816100381610038161003816arg (4 minus 11988821198902)1003816100381610038161003816

1003816lt 120587)

(23)

Case 5 Taking 120601(119905) = E(119888radic119905(119890 minus 119905)) 120595(119905) = 1119905(119890 minus 119905) 119886 = 0and 120574 = 1 in (15) where E(119911) is the complete elliptic integralof the second kind [4 page 35 (33)] defined by

E (119911) = int

1205872

0

radic(1 minus 1199112sin2120579)119889120579 (24)

and using [7 page 280 (3)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1E (119888radic119905 (119890 minus 119905)) 119889119905

=120587

2(119890)

120572+120573minus1

119861 (120572 120573)41198653

[[[[[[

[

minus1

21

2 120572 120573

1198882

1198902

4

1120572 + 120573

2120572 + 120573 + 1

2

]]]]]]

]

(119890Re (120572) Re (120573) gt 010038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(25)

we find the following integral

11986811988912057212057310119890

[E(119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)

]

=

120587

2

(119890)120572+120573minus1

infin

sum

119898=0

(minus12)119898(12)

119898(120572)

119898(120573)

119898(1198882

1198902

4)

119898

(1)119898((120572 + 120573)2)

119898((120572 + 120573 + 1)2)

119898119898

times119903+2119865119905+2

[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 1 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905) 1 minus 120572 minus 119898 1 minus 120573 minus 119898

]]]]

]

(119890Re (120572) Re (120573) gt 0 10038161003816100381610038161003816arg (4 minus 11988821198902)1003816100381610038161003816

1003816lt 120587)

(26)

Case 6 Taking 120601(119905) = arcsin(119888radic119905(119890 minus 119905)) 120595(119905) = 1119905(119890 minus 119905)119886 = 0 and 120574 = 2 in (15) and using [7 page 173 (151)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1arcsin2 (119888radic119905 (119890 minus 119905)) 119889119905

= 1198882

(119890)120572+120573+1

119861 (120572 + 1 120573 + 1)

times51198654

[[[[

[

1 1 1 120572 + 1 120573 + 1

1198882

1198902

43

2 2

120572 + 120573 + 2

2120572 + 120573 + 3

2

]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus110038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(27)

Journal of Mathematics 5

we find the following integral

11986811988912057212057320119890

[arcsin (119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)]

=120572120573119888

2

(119890)120572+120573+1

(120572 + 120573) (120572 + 120573 + 1)

times

infin

sum

119898=0

(1)119898(1)

119898(1)

119898(120572 + 1)

119898(120573 + 1)

119898(1198882

1198902

4)119898

(32)119898(2)

119898((120572 + 120573 + 2)2)

119898((120572 + 120573 + 3)2)

119898119898

times119903+2

119865119905+2

[[[[

[

(119891119903) minus

1

2(1 + 120572 + 120573) minus 119898 minus

1

2(120572 + 120573) minus 119898

4119889

1198902

(119892119905) minus120572 minus 119898 minus120573 minus 119898

]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus110038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(28)

Case 7 Taking 120601(119905) = exp(1198882119905(119890 minus 119905)) erf(119888radic119905(119890 minus 119905)) 120595(119905) =1119905(119890 minus 119905) 119886 = 0 and 120574 = 1 in (15) and using [7 page 185 (1)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1 exp (1198882119905 (119890 minus 119905)) erf (119888radic119905 (119890 minus 119905)) 119889119905

=2

radic120587119888(119890)

120572+120573

119861(120572 +1

2 120573 +

1

2)

times31198653

[[[[[[

[

1 120572 +1

2 120573 +

1

2

1198882

1198902

43

2120572 + 120573 + 1

2120572 + 120573 + 2

2

]]]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus1

2)

(29)

where erf(119911) = (2radic120587)Erf(119911) denotes the error function [4]we find the following integral

11986811988912057212057310119890

[exp (1198882119905 (119890 minus 119905)) erf (119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)

]

=

2

radic120587

119888(119890)120572+120573

119861 (120572 + (12) 120573 + (12))

119861 (120572 120573)

times

infin

sum

119898=0

(1)119898(120572 + (12))

119898(120573 + (12))

119898(1198882

1198902

4)

119898

(32)119898((120572 + 120573 + 1)2)

119898((120572 + 120573 + 2)2)

119898119898

times119903+2119865119905+2

[[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905)

1

2

minus 120572 minus 119898

1

2

minus 120573 minus 119898

]]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus12

)

(30)

Now we establish a theorem and apply it to obtain theevaluations of certain integrals in terms of extended betafunction

3 Integrals in Terms of ExtendedBeta Function

Consider an (119903+1)-variable generating function119866(1199091 1199092

119909119903 119905)which possesses a formal (not necessarily convergent for

119905 = 0) power series expansion in 119905 such that [4 page 80 (8)]

119866 (1199091 1199092 119909

119903 119905) =

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119905119899

(31)

where each member of the generated set 119892119899(1199091 1199092

119909119903)infin

119899=0is independent of 119905 and the coefficient set 119888

119899infin

119899=0may

contain the parameters of the set 119892119899(1199091 1199092 119909

119903)infin

119899=0but is

independent of 1199091 1199092 119909

119903and 119905

Theorem 1 Let the generating function 119866(1199091 1199092 119909

119903 119905)

defined by (31) be such that 119866(1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

remains uniformly convergent for 119906 isin (119886 119888) 120588 120590 ge 0 and120588 + 120590 gt 0 Then

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(32)

where 119861(119909 119910 119887) is the extended beta function defined by (7)

Proof Applying the definition of 119866(1199091 1199092 119909

119903 119905) given in

(31) in the lhs of (32) we get

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119905119899

times int

119888

119886

(119906 minus 119886)120582+120588119899minus1

(119888 minus 119906)120583minus120582+120590119899minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

(33)

which by using (9) yields the rhs of (32)

Corollary 2 With definition (31) and notations as inTheorem 1 one has

int

1

0

119906120582minus1

(1 minus 119906)120583minus120582minus1

119866 (1199091 1199092 119909

119903 119905119906

120588

(1 minus 119906)120590

)

times exp(minus 119887

119906 (1 minus 119906)) 119889119906

6 Journal of Mathematics

=

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(34)

Proof Taking 119886 = 0 and 119888 = 1 in (32) we get (34)

Corollary 3 With definition (31) and notations as inTheorem 1 one has

int

1

minus1

(1 + 119906)120582minus1

(1 minus 119906)120583minus120582minus1

times 119866 (1199091 1199092 119909

119903 119905(1 + 119906)

120588

(1 minus 119906)120590

)

times exp(minus 4119887

(1 minus 1199062)) 119889119906

=

infin

sum

119899=0

(2)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(35)

Proof Taking 119886 = minus1 and 119888 = 1 in (32) we get (35)

Corollary 4 With definition (31) and notations as inTheorem 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)minus120582minus1

119866(1199091 1199092 119909

119903 119905(

119888 minus 119906

119906 minus 119886)

120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=2

(119888 minus 119886) 1198902119887

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903)119870

120582minus120590119899(2119887) 119905

119899

(Re (120582) gt 0Re (119887) gt 0)

(36)

Proof Taking 120583 = 0 and 120588 = minus120590 in (32) and making use ofrelation (10) we get (36)

Now we apply Theorem 1 to derive the evaluations ofcertain Euler type integrals in terms of EBF 119861(120572 120573 119887) Weconsider the following cases

Case 1 Consider the generating function of the multivariableHermite polynomials 119867(12119903)

119899(1199091 1199092 119909

119903) [8 page 602

(2021)]

exp (1199091119905 + 119909

21199052

+ sdot sdot sdot + 119909119903119905119903

)

=

infin

sum

119899=0

119867(12119903)

119899(1199091 1199092 119909

119903)119905119899

119899

(37)

Applying Theorem 1 to generating function (37) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

119903

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(38)

Taking 119903 = 3 in (38) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

3

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(39)

where119867119899(1199091 1199092 1199093) are the 3-variable Hermite polynomials

(3VHP) [9]Again taking 119903 = 2 in (38) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

2

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(1199091 1199092)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(40)

Journal of Mathematics 7

where 119867119899(1199091 1199092) are the 2-variable Hermite-Kampe de-

Feriet polynomials (2VHKdFP) [10]Next replacing 119909

1by 119909 119909

119903by 119910 and taking 119909

2=

1199093

= sdot sdot sdot = 119909119903minus1

= 0 in (38) and using the relation119867(12119903)

119899(119909 0 0 0 119910) = 119892

119903

119899(119909 119910) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

+119910(119905(119906 minus 119886)120588

(119888 minus 119906)120590

)119903

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119892119903

119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(41)

where 119892119903119899(119909 119910) are the Gould-Hopper generalized Hermite

polynomials (GHGHP) [11]Further replacing 119909

1by ]119909 and taking 119909

2= 119909

3= sdot sdot sdot =

119909119903minus1

= 0 119909119903= minus1 in (38) and using the relation119867(12119903)

119899(]119909

0 0 0 minus1) = 119867119899119903](119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+]119909119905(119906 minus 119886)120588(119888 minus 119906)120590minus (119905(119906 minus 119886)120588(119888 minus 119906)120590)119903)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899119903] (119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(42)

where119867119899119903](119909) are the generalizedHermite polynomials [12]

Furthermore taking 119903 = ] = 2 in (42) and using therelation119867

11989922(119909) = 119867

119899(119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+2119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

minus (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)2

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(43)

where119867119899(119909) are the ordinary Hermite polynomials [1]

Case 2 Consider the generating function of the 2-variableLaguerre polynomials 119871

119899(119909 119910) [13]

1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905) =

infin

sum

119899=0

119871119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(44)

which can also be expressed as

exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119899(119909 119910)

119905119899

119899 (45)

where 1198620(119909) denotes the 0th-order Tricomi function [4]

Applying Theorem 1 to generating functions (44) and(45) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(46)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(47)

respectively

8 Journal of Mathematics

Taking 119910 = 1 in (46) and (47) and using the relation119871119899(119909 1) = 119871

119899(119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(48)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(49)

respectively where 119871119899(119909) are the ordinary Laguerre polyno-

mials [1]Again taking119910 = 0 in (46) and (47) and using the relation

119871119899(119909 0) = (minus119909)

119899

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(50)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(51)respectively

Next taking 119909 = 0 in (46) and (47) and using the relation119871119899(0 119910) = 119910

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590exp(minus 119887(119888 minus 119886)

2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) (119910119905)119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(52)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(119910119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(53)

respectively Replacing 119910 by minus119909 in (53) it reduces to (50)

Case 3 Consider the generating function of the Hermite-Appell polynomials

119867119860119899(1199091 1199092 1199093) [14 page 759 (23)]

119860 (119905) exp (1199091119905 + 119909

21199052

+ 11990931199053

) =

infin

sum

119899=0

119867119860119899(1199091 1199092 1199093)119905119899

119899

(54)

Applying Theorem 1 to generating function (54) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+

3

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119860119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(55)

Journal of Mathematics 9

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (55) and

using the relation119867119860119899(119909 0 0) = 119860

119899(119909) [14 page 760 (26)]

we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(56)

where 119860119899(119909) are the Appell polynomials [4 15]

Case 4 Consider the generating function of the Laguerre-Appell polynomials

119871119860119899(119909 119910) [16]

119860 (119905) [1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905)]

=

infin

sum

119899=0

119871119860119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(57)

which can also be expressed as follows

119860 (119905) exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119860119899(119909 119910)

119905119899

119899 (58)

Applying Theorem 1 to generating functions (57) and(58) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(59)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(60)

respectivelyTaking 119910 = 0 and replacing 119909 by minus119909 in (59) and (60) and

using the relation119871119860119899(119909 0) = 119860

119899(minus119909)119899 [16 page 9 (28)]

we get (56) and

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(61)

respectively

Case 5 Consider the generating function of the Hermite-Sheffer polynomials

119867119904119899(1199091 1199092 1199093) [17]

119860 (119905) exp (1199091119867(119905) + 119909

21198672

(119905) + 11990931198673

(119905))

=

infin

sum

119899=0

119867119904119899(1199091 1199092 1199093)119905119899

119899

(62)

Applying Theorem 1 to generating function (62) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

3

sum

119895=1

119909119895119867119895

(119905(119906 minus 119886)120588

(119888 minus 119906)120590

))119889119906

10 Journal of Mathematics

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119904119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(63)

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (63) and

using the relation119867119904119899(119909 0 0) = 119904

119899(119909) [17 page 16 (23)] we

get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119867 (119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119904119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(64)

where 119904119899(119909) are the Sheffer polynomials [15 18 19]

4 Concluding Remarks

In this paper we have derived the evaluations of certain Eulertype integralsWehave also established a theoremand appliedit to obtain evaluations of some integrals in terms of EBF119861(120572 120573 119887) We remark that these results can be extended tomultivariable case

To give an example we consider the following integralrepresentation [5 page 965 (20)] of Lauricellarsquos multiplehypergeometric series 119865(119899)

119863[4]

int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

119889119905

= 119861 (120572 120573) 119865(119899)

119863[120572 120572

1 1205722 120572

119899 120572 + 120573 119909

1 1199092 119909

119899]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(65)

Now taking 120601(119905) = prod119899

119894=1(1 minus 119909

119894119905)minus120572119894 120595(119905) = 1119905(1 minus 119905)

119886 = 0 and 119890 = 120574 = 1 in (15) using integral (65) in the rhs

after expanding119903119865119905[

(119891119903)

119889120595(119905)

(119892119905)

] in series and simplifying weget

119868119889120572120573101

[

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

1

119905 (1 minus 119905)]

=

infin

sum

11989811198982119898119899=0

(120572)1198981+1198982+sdotsdotsdot+119898

119899

(1205721)1198981

(1205722)1198982

sdot sdot sdot (120572119899)119898119899

(120572 + 120573)1198981+1198982+sdotsdotsdot+119898

119899

1199091198981

1

1198981

1199091198982

2

1198982sdot sdot sdot

119909119898119899

119899

119898119899

times119903+2119865119905+2

[[[[[[[

[

(119891119903) 1

2(1 minus 120572 minus 120573 minus

119899

sum

119894=1

119898119894) 1 minus

1

2(120572 + 120573 +

119899

sum

119894=1

119898119894)

4119889

(119892119905) 1 minus (120572 +

119899

sum

119894=1

119898119894) 1 minus 120573

]]]]]]]

]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(66)

We note that for 1205723= 120572

4= sdot sdot sdot = 0 (66) reduces to (17)

Further we extendTheorem 1 as follows

Theorem 5 Let the conditions for 119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

11989911989811198982119898119904=0

(119888 minus 119886)120583+(120588+120590)119899+119898

1+1198982+sdotsdotsdot+119898

119904minus1

times 119888119899119892119899(1199091 1199092 119909

119903)

times 119861(120582 + 120588119899 +

119904

sum

119894=1

119898119894 120583 minus 120582 + 120590119899 119887)

119904

prod

119894=1

(120572119894)119898119894

119910119898119894

119894

119898119894

119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(67)

Journal of Mathematics 11

The proof of this theorem is similar to that of Theorem 1The following are the consequences of Theorem 5

Corollary 6 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and119904 = 2 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

2

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899)

times 1198651[120582 + 120588119899 120572

1 1205722 120583 + (120588 + 120590) 119899

(119888 minus 119886) 1199101 (119888 minus 119886) 119910

2] 119905119899

(Re (120583) gt Re (120582) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(68)

Corollary 7 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and120588 = 120590 = 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905 (119906 minus 119886) (119888 minus 119906)) 119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904] 119905119899

(Re (120583) gt Re (120582) gt 0)

(69)

Example 8 Applying Corollary 7 to generating function (37)of the multivariable Hermite polynomials 119867(12119903)

119899(1199091 1199092

119909119903) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times exp(119903

sum

119895=1

119909119895(119905 (119906 minus 119886) (119888 minus 119906))

119895

)119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904]119905119899

119899

(Re (120583) gt Re (120582) gt 0)

(70)Remark 9 Taking 119886 = 0 and 119888 = 1 in (67) we get an extensionof integral (34)

Remark 10 Taking 119886 = minus1 and 119888 = 1 in (67) we get anextension of integral (35)

Clearly it appears that by applying Theorem 5 andCorollaries 6 and 7 to the generating functions defined by(37) (44) (45) (54) (57) (58) and (62) and also to someother generating functions a number of interesting integralscan be obtained

Further it is remarked that the operational methodsprovide useful tools to estimate specific theorems in variousfields of analysisThis approach requires the validation of anyof its consequences by a more rigorous procedure This isindeed the case of the Ramanujan master theorem [20 21]An operational method already employed to formulate ageneralization of the Ramanujan master theorem is appliedfor the evaluation of certain integrals by Babusci et al [22]

The technique adopted in [22] provides a very flexible andpowerful tool yielding new results encompassing differentaspects of the theory of special functions The possibility ofusing the method outlined in [22] for the integrals evaluatedin this article is a further research problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to thereviewer(s) for helpful suggestions and comments towardsthe improvement of this paper

References

[1] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985

[2] M A Chaudhry and S M Zubair ldquoGeneralized incompletegamma functions with applicationsrdquo Journal of Computationaland Applied Mathematics vol 55 no 1 pp 99ndash124 1994

[3] M A Chaudhry A Qadir M Rafique and S M ZubairldquoExtension of Eulerrsquos beta functionrdquo Journal of Computationaland Applied Mathematics vol 78 no 1 pp 19ndash32 1997

12 Journal of Mathematics

[4] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions John Wiley amp Sons New York NY USA 1984

[5] M E H Ismail and J Pitman ldquoAlgebraic evaluations of someEuler integrals duplication formulae for Appellrsquos hypergeomet-ric function 119865

1 and Brownian variationsrdquo Canadian Journal of

Mathematics vol 52 no 5 pp 961ndash981 2000[6] J Pitman ldquoBrownian motion bridge excursion and meander

characterized by sampling at independent uniform timesrdquoElectronic Journal of Probability vol 11 no 4 pp 1ndash33 1999

[7] Y A Brychkov Handbook of Special Functions DerivativesIntegrals Series and Other Formulas CRC Press New York NYUSA 2008

[8] G Dattoli S Lorenzutta G Maino A Torre and C CesaranoldquoGeneralized Hermite polynomials and super-Gaussian formsrdquoJournal of Mathematical Analysis and Applications vol 203 no3 pp 597ndash609 1996

[9] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[10] P Appell and J K de Feriet Fonctions Hypergeometriques etHyperspheriques Polynomes drsquoHermite Gauthier-Villars ParisFarnce 1926

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962

[12] M Lahiri ldquoOn a generalisation of Hermite polynomialsrdquoProceedings of the American Mathematical Society vol 27 pp117ndash121 1971

[13] G Dattoli and A Torre ldquoOperatorial methods and two variableLaguerre polynomialsrdquo Atti della Accademia delle Scienze diTorino Classe di Scienze Fisiche Matematiche e Naturali vol132 pp 1ndash7 1998

[14] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009

[15] S RomanTheUmbral Calculus Academic Press NewYorkNYUSA 1984

[16] S Khan M W Al-Saad and R Khan ldquoLaguerre-based Appellpolynomials properties and applicationsrdquo Mathematical andComputer Modelling vol 52 no 1-2 pp 247ndash259 2010

[17] S Khan M W Al-Saad and G Yasmin ldquoSome properties ofHermite-based Sheffer polynomialsrdquo Applied Mathematics andComputation vol 217 no 5 pp 2169ndash2183 2010

[18] E D Rainville Special Functions Macmillan New York NYUSA 1960

[19] E D Rainville SpecialFunctions Chelsea Publishing NewYorkNY USA 1971

[20] B C Berndt Ramanujanrsquos NotebooksmdashPart I Springer NewYork NY USA 1985

[21] G H Hardy Ramanujan Twelve Lectures on Subjects Suggestedby His Life and Work Cambridge University Press CambridgeUK 1940

[22] D Babusci G Dattoli G H E Duchamp K Gorska and K APenson ldquoDefinite integrals and operational methodsrdquo AppliedMathematics and Computation vol 219 no 6 pp 3017ndash30212012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Mathematics

In viewof the fact that the regularization factor is useful inextending the domain of the gamma and zeta functions Thedomain of other special functions could be usefully extendedin a similar manner

In particular Eulerrsquos beta function 119861(120572 120573) has a closerelationship to gamma function that is

119861 (120572 120573) = 119861 (120573 120572) =Γ (120572) Γ (120573)

Γ (120572 + 120573) (5)

and can be usefully extendable We first recall the basic Eulerintegral which defines the beta function as follows

119861 (120572 120573) = int

1

0

119906120572minus1

(1 minus 119906)120573minus1

119889119906 (Re (120572) Re (120573) gt 0)

(6)

Keeping in view the fact that for the 120572-120573 symmetry to bepreserved there must be symmetry of the integrand in 119906 and1 minus 119906 Chaudhry et al [3] introduced an extension of Eulerrsquosbeta function 119861(120572 120573) in the following form

119861 (120572 120573 119887) = int

1

0

119905120572minus1

(1 minus 119905)120573minus1 exp(minus 119887

119905 (1 minus 119905)) 119889119905

(Re (119887) gt 0)

(7)

which for 119887 = 0 reduces to the original beta functionThe transformations 119905 = (119906 + 1)2 and 119905 = (119906 minus 119886)

(119888 minus 119886) in (7) yield the following integral representations forthe extended beta function (EBF) 119861(120572 120573 119887) [3 page 22 (29)(211)]

119861 (120572 120573 119887)

= 21minus120572minus120573

times int

1

minus1

(1 + 119906)120572minus1

(1 minus 119906)120573minus1 exp(minus 4119887

(1 minus 1199062)) 119889119906

(Re (119887) gt 0)

(8)

119861 (120572 120573 119887)

= (119888 minus 119886)1minus120572minus120573

times int

119888

119886

(119906 minus 119886)120572minus1

(119888 minus 119906)120573minus1 exp(minus 119887(119888 minus 119886)

2

(119906 minus 119886) (119888 minus 119906)) 119889119906

(Re (119887) gt 0)

(9)

respectivelyThe EBF 119861(120572 120573 119887) is extremely useful in the sense that

most of the properties of the beta function carry overnaturally and simply for it This extension is also importantdue to the fact that this function is related to other specialfunctions for particular values of the variables

We note the following connections [3]

119861 (120572 minus120572 119887) = 2119890minus2119887

119870120572(2119887) (Re (119887) gt 0) (10)

119861 (120572 120572 119887)

= radic1205872minus120572

119887(120572minus1)2

119890minus2119887

119882minus12057221205722

(4119887)

= radic12058721minus2120572

119890minus4119887

Ψ(1

2 1 minus 120572 4119887) (Re (119887) gt 0)

(11)

where 119882119898119899

(119911) denotes the Whittaker function [1] andΨ(119886 119888 119911) denotes the second form of solutions of Kummerrsquosequation [4]

Also we note that

119861 (0 0 119887) = 2119890minus2119887

1198700(2119887) (Re (119887) gt 0)

119861 (1

2 minus1

2 119887) = radic

120587

119887119890minus4119887

(Re (119887) gt 0)

119861 (1

21

2 119887) = 120587Erfc (2radic119887) (Re (119887) gt 0)

(12)

where Erfc(119911) = (radic1205872)erfc(119911) denotes the complementaryerror function [4]

Explicit evaluations of some integrals of Euler type

int

1

0

119906120572minus1

(1 minus 119906)120573minus1

119891 (119906) 119889119906 (13)

for some particular functions 119891 specially in the symmetriccase 120572 = 120573 are derived in [5] These evaluations are relatedto various reduction formulae for hypergeometric functionsrepresented by such integrals These formulae generalize theevaluations of some symmetric Euler integrals implied by thefollowing result due to Pitman [6]

If a standard Brownian bridge is sampled at time 0time 1 and at 119899 independent random times with uniformdistribution on [0 1] then the broken line approximationto the bridge obtained from these 119899 + 2 values has a totalvariation whose mean square is 119899(119899 + 1)(2119899 + 1)

Motivated and inspired by the work of Ismail and Pitman[5] and Chaudhry et al [3] in this paper we derive theevaluations of certain Euler type integrals and some integralsin terms of EBF 119861(120572 120573 119887) In Section 2 we obtain theevaluations of certain integrals of the following type

119868119889120572120573120574119886119890

[120601 (119905) 120595 (119905)]

=1

119861 (120572 120573)int

119890

119886

(119905 minus 119886)120572minus1

(119890 minus 119905)120573minus1

(120601 (119905))120574

times119903119865119905

[

[

(119891119903)

119889120595 (119905)

(119892119905)

]

]

119889119905

(14)

where119903119865119905denotes the generalized hypergeometric series [4]

for particular functions 120601(119905) and 120595(119905) In Section 3 weestablish a theorem on EBF 119861(120572 120573 119887) and apply it to obtainevaluations of certain integrals in terms of EBF 119861(120572 120573 119887)Finally we give some concluding remarks in Section 4

Journal of Mathematics 3

2 Euler Type Integrals

We derive the evaluations of certain integrals of the followingtype

119868119889120572120573120574119886119890

[120601 (119905) 120595 (119905)]

=1

119861 (120572 120573)int

119890

119886

(119905 minus 119886)120572minus1

(119890 minus 119905)120573minus1

(120601 (119905))120574

times119903119865119905

[

[

(119891119903)

119889120595 (119905)

(119892119905)

]

]

119889119905

(15)

We evaluate the integrals of type (15) for particularfunctions 120601(119905) and 120595(119905) by considering the following cases

Case 1 Taking120601(119905) = (1minus1199091119905)minus1205721(1minus119909

2119905)minus1205722 120595(119905) = 1119905(1minus119905)

119886 = 0 and 119890 = 120574 = 1 in (15) and using [5 page 962 (7)]

int

1

0

119905120572minus1

(1 minus 119905)120573minus1

(1 minus 1199091119905)minus1205721

(1 minus 1199092119905)minus1205722

119889119905

= 119861 (120572 120573) 1198651[120572 120572

1 120572

2 120572 + 120573 119909

1 119909

2]

(Re (120572) Re (120573) gt 0 max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 lt 1)

(16)

where 1198651denotes the Appell function [4] in the rhs

after expanding119903119865119905[

(119891119903)

119889120595(119905)

(119892119905)

] and simplifying we find the

following integral

119868119889120572120573101

[(1 minus 1199091119905)minus1205721

(1 minus 1199092119905)minus1205722

1

119905 (1 minus 119905)]

=

infin

sum

11989811198982=0

(120572)1198981+1198982

(1205721)1198981

(1205722)1198982

1199091198981

11199091198982

2

(120572 + 120573)1198981+1198982

1198981119898

2

times119903+2119865119905+2

[[

[

(119891119903) 1

2(1 minus 120572 minus 120573 minus 119898

1minus 119898

2) 1 minus

1

2(120572 + 120573 + 119898

1+ 119898

2)

4119889

(119892119905) 1 minus (120572 + 119898

1+ 119898

2) 1 minus 120573

]]

]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 lt 1)

(17)

Case 2 Taking 120601(119905) =119901

119865119902[

(119886119901)

119888119905(119890minus119905)

(119887119902)

] 120595(119905) = 1119905(119890 minus 119905)

119886 = 0 and 120574 = 1 in (15) and using [7 page 303 (1)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1

119901119865119902

[[

[

(119886119901)

119888119905 (119890 minus 119905)

(119887119902)

]]

]

119889119905

= (119890)120572+120573minus1

119861 (120572 120573)

times119901+2

119865119902+2

[[[[[

[

(119886119901) 120572 120573

1198881198902

4

(119887119902)

120572 + 120573

2120572 + 120573 + 1

2

]]]]]

]

(119890Re (120572) Re (120573) gt 010038161003816100381610038161003816arg (4 minus 1198881198902)10038161003816100381610038161003816 lt 120587)

(18)

we find the following integral

11986811988912057212057310119890

[

[

119901119865119902

[

[

(119886119901)

119888119905 (119890 minus 119905)

(119887119902)

]

]

1

119905 (119890 minus 119905)

]

]

= (119890)120572+120573minus1

infin

sum

119898=0

(119886119901)119898

(120572)119898(120573)

119898(119888119890

2

4)

119898

(119887119902)119898

((120572 + 120573)2)119898((120572 + 120573 + 1)2)

119898119898

times119903+2119865119905+2

[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 1 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905) 1 minus 120572 minus 119898 1 minus 120573 minus 119898

]]]]

]

(119890Re (120572) Re (120573) gt 0 10038161003816100381610038161003816arg (4 minus 1198881198902)1003816100381610038161003816

1003816lt 120587)

(19)

Case 3 Taking120601(119905) =119901

119865119902[

(119886119901)

119888119905(119890minus119905)

(119887119902)

]120595(119905) = 1(1199052

(119890 minus 119905)2

)

119886 = 0 and 120574 = 1 and proceeding as in Case 2 we find thefollowing integral

11986811988912057212057310119890

[[

[

119901119865119902

[[

[

(119886119901)

119888119905 (119890 minus 119905)

(119887119902)

]]

]

1

1199052(119890 minus 119905)2

]]

]

= (119890)120572+120573minus1

infin

sum

119898=0

(119886119901)119898

(120572)119898(120573)

119898(119888119890

2

4)119898

(119887119902)119898

((120572 + 120573) 2)119898((120572 + 120573 + 1) 2)

119898119898

4 Journal of Mathematics

times119903+6119865119905+8

[[[[[[

[

(119891119903) 119895 minus 120572 minus 120573

21

2(1 minus 120572 minus 120573) minus 119898 1 minus

1

2(120572 + 120573) minus 119898

4119889

1198904

(119892119905) 119895 minus 120572 minus 120573

41 minus 120572 minus 119898

22 minus 120572 minus 119898

21 minus 120573 minus 119898

22 minus 120573 minus 119898

2

]]]]]]

]

(119895 = 1 2 3 4) (119890Re (120572) Re (120573) gt 010038161003816100381610038161003816arg (4 minus 1198881198902)10038161003816100381610038161003816 lt 120587)

(20)

Case 4 Taking 120601(119905) = K(119888radic119905(119890 minus 119905)) 120595(119905) = 1119905(119890 minus 119905) 119886 = 0and 120574 = 1 in (15) where K(119911) is the complete elliptic integralof the first kind [4 page 35 (32)] defined by

K (119911) = int

1205872

0

119889120579

radic(1 minus 1199112sin2120579)(21)

and using [7 page 265 (4)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1K (119888radic119905 (119890 minus 119905)) 119889119905

=120587

2(119890)

120572+120573minus1

119861 (120572 120573)41198653

[[[[[[

[

1

21

2 120572 120573

1198882

1198902

4

1120572 + 120573

2120572 + 120573 + 1

2

]]]]]]

]

(Re (120572) Re (120573) gt 010038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(22)

we find the following integral

11986811988912057212057310119890

[K(119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)

]

=

120587

2

(119890)120572+120573minus1

infin

sum

119898=0

(12)119898(12)

119898(120572)

119898(120573)

119898(1198882

1198902

4)

119898

(1)119898((120572 + 120573)2)

119898((120572 + 120573 + 1)2)

119898119898

times119903+2119865119905+2

[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 1 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905) 1 minus 120572 minus 119898 1 minus 120573 minus 119898

]]]]

]

(Re (120572) Re (120573) gt 0 10038161003816100381610038161003816arg (4 minus 11988821198902)1003816100381610038161003816

1003816lt 120587)

(23)

Case 5 Taking 120601(119905) = E(119888radic119905(119890 minus 119905)) 120595(119905) = 1119905(119890 minus 119905) 119886 = 0and 120574 = 1 in (15) where E(119911) is the complete elliptic integralof the second kind [4 page 35 (33)] defined by

E (119911) = int

1205872

0

radic(1 minus 1199112sin2120579)119889120579 (24)

and using [7 page 280 (3)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1E (119888radic119905 (119890 minus 119905)) 119889119905

=120587

2(119890)

120572+120573minus1

119861 (120572 120573)41198653

[[[[[[

[

minus1

21

2 120572 120573

1198882

1198902

4

1120572 + 120573

2120572 + 120573 + 1

2

]]]]]]

]

(119890Re (120572) Re (120573) gt 010038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(25)

we find the following integral

11986811988912057212057310119890

[E(119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)

]

=

120587

2

(119890)120572+120573minus1

infin

sum

119898=0

(minus12)119898(12)

119898(120572)

119898(120573)

119898(1198882

1198902

4)

119898

(1)119898((120572 + 120573)2)

119898((120572 + 120573 + 1)2)

119898119898

times119903+2119865119905+2

[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 1 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905) 1 minus 120572 minus 119898 1 minus 120573 minus 119898

]]]]

]

(119890Re (120572) Re (120573) gt 0 10038161003816100381610038161003816arg (4 minus 11988821198902)1003816100381610038161003816

1003816lt 120587)

(26)

Case 6 Taking 120601(119905) = arcsin(119888radic119905(119890 minus 119905)) 120595(119905) = 1119905(119890 minus 119905)119886 = 0 and 120574 = 2 in (15) and using [7 page 173 (151)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1arcsin2 (119888radic119905 (119890 minus 119905)) 119889119905

= 1198882

(119890)120572+120573+1

119861 (120572 + 1 120573 + 1)

times51198654

[[[[

[

1 1 1 120572 + 1 120573 + 1

1198882

1198902

43

2 2

120572 + 120573 + 2

2120572 + 120573 + 3

2

]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus110038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(27)

Journal of Mathematics 5

we find the following integral

11986811988912057212057320119890

[arcsin (119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)]

=120572120573119888

2

(119890)120572+120573+1

(120572 + 120573) (120572 + 120573 + 1)

times

infin

sum

119898=0

(1)119898(1)

119898(1)

119898(120572 + 1)

119898(120573 + 1)

119898(1198882

1198902

4)119898

(32)119898(2)

119898((120572 + 120573 + 2)2)

119898((120572 + 120573 + 3)2)

119898119898

times119903+2

119865119905+2

[[[[

[

(119891119903) minus

1

2(1 + 120572 + 120573) minus 119898 minus

1

2(120572 + 120573) minus 119898

4119889

1198902

(119892119905) minus120572 minus 119898 minus120573 minus 119898

]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus110038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(28)

Case 7 Taking 120601(119905) = exp(1198882119905(119890 minus 119905)) erf(119888radic119905(119890 minus 119905)) 120595(119905) =1119905(119890 minus 119905) 119886 = 0 and 120574 = 1 in (15) and using [7 page 185 (1)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1 exp (1198882119905 (119890 minus 119905)) erf (119888radic119905 (119890 minus 119905)) 119889119905

=2

radic120587119888(119890)

120572+120573

119861(120572 +1

2 120573 +

1

2)

times31198653

[[[[[[

[

1 120572 +1

2 120573 +

1

2

1198882

1198902

43

2120572 + 120573 + 1

2120572 + 120573 + 2

2

]]]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus1

2)

(29)

where erf(119911) = (2radic120587)Erf(119911) denotes the error function [4]we find the following integral

11986811988912057212057310119890

[exp (1198882119905 (119890 minus 119905)) erf (119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)

]

=

2

radic120587

119888(119890)120572+120573

119861 (120572 + (12) 120573 + (12))

119861 (120572 120573)

times

infin

sum

119898=0

(1)119898(120572 + (12))

119898(120573 + (12))

119898(1198882

1198902

4)

119898

(32)119898((120572 + 120573 + 1)2)

119898((120572 + 120573 + 2)2)

119898119898

times119903+2119865119905+2

[[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905)

1

2

minus 120572 minus 119898

1

2

minus 120573 minus 119898

]]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus12

)

(30)

Now we establish a theorem and apply it to obtain theevaluations of certain integrals in terms of extended betafunction

3 Integrals in Terms of ExtendedBeta Function

Consider an (119903+1)-variable generating function119866(1199091 1199092

119909119903 119905)which possesses a formal (not necessarily convergent for

119905 = 0) power series expansion in 119905 such that [4 page 80 (8)]

119866 (1199091 1199092 119909

119903 119905) =

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119905119899

(31)

where each member of the generated set 119892119899(1199091 1199092

119909119903)infin

119899=0is independent of 119905 and the coefficient set 119888

119899infin

119899=0may

contain the parameters of the set 119892119899(1199091 1199092 119909

119903)infin

119899=0but is

independent of 1199091 1199092 119909

119903and 119905

Theorem 1 Let the generating function 119866(1199091 1199092 119909

119903 119905)

defined by (31) be such that 119866(1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

remains uniformly convergent for 119906 isin (119886 119888) 120588 120590 ge 0 and120588 + 120590 gt 0 Then

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(32)

where 119861(119909 119910 119887) is the extended beta function defined by (7)

Proof Applying the definition of 119866(1199091 1199092 119909

119903 119905) given in

(31) in the lhs of (32) we get

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119905119899

times int

119888

119886

(119906 minus 119886)120582+120588119899minus1

(119888 minus 119906)120583minus120582+120590119899minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

(33)

which by using (9) yields the rhs of (32)

Corollary 2 With definition (31) and notations as inTheorem 1 one has

int

1

0

119906120582minus1

(1 minus 119906)120583minus120582minus1

119866 (1199091 1199092 119909

119903 119905119906

120588

(1 minus 119906)120590

)

times exp(minus 119887

119906 (1 minus 119906)) 119889119906

6 Journal of Mathematics

=

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(34)

Proof Taking 119886 = 0 and 119888 = 1 in (32) we get (34)

Corollary 3 With definition (31) and notations as inTheorem 1 one has

int

1

minus1

(1 + 119906)120582minus1

(1 minus 119906)120583minus120582minus1

times 119866 (1199091 1199092 119909

119903 119905(1 + 119906)

120588

(1 minus 119906)120590

)

times exp(minus 4119887

(1 minus 1199062)) 119889119906

=

infin

sum

119899=0

(2)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(35)

Proof Taking 119886 = minus1 and 119888 = 1 in (32) we get (35)

Corollary 4 With definition (31) and notations as inTheorem 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)minus120582minus1

119866(1199091 1199092 119909

119903 119905(

119888 minus 119906

119906 minus 119886)

120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=2

(119888 minus 119886) 1198902119887

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903)119870

120582minus120590119899(2119887) 119905

119899

(Re (120582) gt 0Re (119887) gt 0)

(36)

Proof Taking 120583 = 0 and 120588 = minus120590 in (32) and making use ofrelation (10) we get (36)

Now we apply Theorem 1 to derive the evaluations ofcertain Euler type integrals in terms of EBF 119861(120572 120573 119887) Weconsider the following cases

Case 1 Consider the generating function of the multivariableHermite polynomials 119867(12119903)

119899(1199091 1199092 119909

119903) [8 page 602

(2021)]

exp (1199091119905 + 119909

21199052

+ sdot sdot sdot + 119909119903119905119903

)

=

infin

sum

119899=0

119867(12119903)

119899(1199091 1199092 119909

119903)119905119899

119899

(37)

Applying Theorem 1 to generating function (37) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

119903

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(38)

Taking 119903 = 3 in (38) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

3

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(39)

where119867119899(1199091 1199092 1199093) are the 3-variable Hermite polynomials

(3VHP) [9]Again taking 119903 = 2 in (38) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

2

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(1199091 1199092)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(40)

Journal of Mathematics 7

where 119867119899(1199091 1199092) are the 2-variable Hermite-Kampe de-

Feriet polynomials (2VHKdFP) [10]Next replacing 119909

1by 119909 119909

119903by 119910 and taking 119909

2=

1199093

= sdot sdot sdot = 119909119903minus1

= 0 in (38) and using the relation119867(12119903)

119899(119909 0 0 0 119910) = 119892

119903

119899(119909 119910) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

+119910(119905(119906 minus 119886)120588

(119888 minus 119906)120590

)119903

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119892119903

119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(41)

where 119892119903119899(119909 119910) are the Gould-Hopper generalized Hermite

polynomials (GHGHP) [11]Further replacing 119909

1by ]119909 and taking 119909

2= 119909

3= sdot sdot sdot =

119909119903minus1

= 0 119909119903= minus1 in (38) and using the relation119867(12119903)

119899(]119909

0 0 0 minus1) = 119867119899119903](119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+]119909119905(119906 minus 119886)120588(119888 minus 119906)120590minus (119905(119906 minus 119886)120588(119888 minus 119906)120590)119903)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899119903] (119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(42)

where119867119899119903](119909) are the generalizedHermite polynomials [12]

Furthermore taking 119903 = ] = 2 in (42) and using therelation119867

11989922(119909) = 119867

119899(119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+2119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

minus (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)2

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(43)

where119867119899(119909) are the ordinary Hermite polynomials [1]

Case 2 Consider the generating function of the 2-variableLaguerre polynomials 119871

119899(119909 119910) [13]

1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905) =

infin

sum

119899=0

119871119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(44)

which can also be expressed as

exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119899(119909 119910)

119905119899

119899 (45)

where 1198620(119909) denotes the 0th-order Tricomi function [4]

Applying Theorem 1 to generating functions (44) and(45) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(46)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(47)

respectively

8 Journal of Mathematics

Taking 119910 = 1 in (46) and (47) and using the relation119871119899(119909 1) = 119871

119899(119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(48)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(49)

respectively where 119871119899(119909) are the ordinary Laguerre polyno-

mials [1]Again taking119910 = 0 in (46) and (47) and using the relation

119871119899(119909 0) = (minus119909)

119899

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(50)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(51)respectively

Next taking 119909 = 0 in (46) and (47) and using the relation119871119899(0 119910) = 119910

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590exp(minus 119887(119888 minus 119886)

2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) (119910119905)119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(52)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(119910119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(53)

respectively Replacing 119910 by minus119909 in (53) it reduces to (50)

Case 3 Consider the generating function of the Hermite-Appell polynomials

119867119860119899(1199091 1199092 1199093) [14 page 759 (23)]

119860 (119905) exp (1199091119905 + 119909

21199052

+ 11990931199053

) =

infin

sum

119899=0

119867119860119899(1199091 1199092 1199093)119905119899

119899

(54)

Applying Theorem 1 to generating function (54) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+

3

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119860119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(55)

Journal of Mathematics 9

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (55) and

using the relation119867119860119899(119909 0 0) = 119860

119899(119909) [14 page 760 (26)]

we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(56)

where 119860119899(119909) are the Appell polynomials [4 15]

Case 4 Consider the generating function of the Laguerre-Appell polynomials

119871119860119899(119909 119910) [16]

119860 (119905) [1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905)]

=

infin

sum

119899=0

119871119860119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(57)

which can also be expressed as follows

119860 (119905) exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119860119899(119909 119910)

119905119899

119899 (58)

Applying Theorem 1 to generating functions (57) and(58) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(59)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(60)

respectivelyTaking 119910 = 0 and replacing 119909 by minus119909 in (59) and (60) and

using the relation119871119860119899(119909 0) = 119860

119899(minus119909)119899 [16 page 9 (28)]

we get (56) and

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(61)

respectively

Case 5 Consider the generating function of the Hermite-Sheffer polynomials

119867119904119899(1199091 1199092 1199093) [17]

119860 (119905) exp (1199091119867(119905) + 119909

21198672

(119905) + 11990931198673

(119905))

=

infin

sum

119899=0

119867119904119899(1199091 1199092 1199093)119905119899

119899

(62)

Applying Theorem 1 to generating function (62) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

3

sum

119895=1

119909119895119867119895

(119905(119906 minus 119886)120588

(119888 minus 119906)120590

))119889119906

10 Journal of Mathematics

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119904119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(63)

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (63) and

using the relation119867119904119899(119909 0 0) = 119904

119899(119909) [17 page 16 (23)] we

get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119867 (119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119904119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(64)

where 119904119899(119909) are the Sheffer polynomials [15 18 19]

4 Concluding Remarks

In this paper we have derived the evaluations of certain Eulertype integralsWehave also established a theoremand appliedit to obtain evaluations of some integrals in terms of EBF119861(120572 120573 119887) We remark that these results can be extended tomultivariable case

To give an example we consider the following integralrepresentation [5 page 965 (20)] of Lauricellarsquos multiplehypergeometric series 119865(119899)

119863[4]

int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

119889119905

= 119861 (120572 120573) 119865(119899)

119863[120572 120572

1 1205722 120572

119899 120572 + 120573 119909

1 1199092 119909

119899]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(65)

Now taking 120601(119905) = prod119899

119894=1(1 minus 119909

119894119905)minus120572119894 120595(119905) = 1119905(1 minus 119905)

119886 = 0 and 119890 = 120574 = 1 in (15) using integral (65) in the rhs

after expanding119903119865119905[

(119891119903)

119889120595(119905)

(119892119905)

] in series and simplifying weget

119868119889120572120573101

[

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

1

119905 (1 minus 119905)]

=

infin

sum

11989811198982119898119899=0

(120572)1198981+1198982+sdotsdotsdot+119898

119899

(1205721)1198981

(1205722)1198982

sdot sdot sdot (120572119899)119898119899

(120572 + 120573)1198981+1198982+sdotsdotsdot+119898

119899

1199091198981

1

1198981

1199091198982

2

1198982sdot sdot sdot

119909119898119899

119899

119898119899

times119903+2119865119905+2

[[[[[[[

[

(119891119903) 1

2(1 minus 120572 minus 120573 minus

119899

sum

119894=1

119898119894) 1 minus

1

2(120572 + 120573 +

119899

sum

119894=1

119898119894)

4119889

(119892119905) 1 minus (120572 +

119899

sum

119894=1

119898119894) 1 minus 120573

]]]]]]]

]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(66)

We note that for 1205723= 120572

4= sdot sdot sdot = 0 (66) reduces to (17)

Further we extendTheorem 1 as follows

Theorem 5 Let the conditions for 119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

11989911989811198982119898119904=0

(119888 minus 119886)120583+(120588+120590)119899+119898

1+1198982+sdotsdotsdot+119898

119904minus1

times 119888119899119892119899(1199091 1199092 119909

119903)

times 119861(120582 + 120588119899 +

119904

sum

119894=1

119898119894 120583 minus 120582 + 120590119899 119887)

119904

prod

119894=1

(120572119894)119898119894

119910119898119894

119894

119898119894

119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(67)

Journal of Mathematics 11

The proof of this theorem is similar to that of Theorem 1The following are the consequences of Theorem 5

Corollary 6 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and119904 = 2 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

2

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899)

times 1198651[120582 + 120588119899 120572

1 1205722 120583 + (120588 + 120590) 119899

(119888 minus 119886) 1199101 (119888 minus 119886) 119910

2] 119905119899

(Re (120583) gt Re (120582) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(68)

Corollary 7 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and120588 = 120590 = 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905 (119906 minus 119886) (119888 minus 119906)) 119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904] 119905119899

(Re (120583) gt Re (120582) gt 0)

(69)

Example 8 Applying Corollary 7 to generating function (37)of the multivariable Hermite polynomials 119867(12119903)

119899(1199091 1199092

119909119903) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times exp(119903

sum

119895=1

119909119895(119905 (119906 minus 119886) (119888 minus 119906))

119895

)119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904]119905119899

119899

(Re (120583) gt Re (120582) gt 0)

(70)Remark 9 Taking 119886 = 0 and 119888 = 1 in (67) we get an extensionof integral (34)

Remark 10 Taking 119886 = minus1 and 119888 = 1 in (67) we get anextension of integral (35)

Clearly it appears that by applying Theorem 5 andCorollaries 6 and 7 to the generating functions defined by(37) (44) (45) (54) (57) (58) and (62) and also to someother generating functions a number of interesting integralscan be obtained

Further it is remarked that the operational methodsprovide useful tools to estimate specific theorems in variousfields of analysisThis approach requires the validation of anyof its consequences by a more rigorous procedure This isindeed the case of the Ramanujan master theorem [20 21]An operational method already employed to formulate ageneralization of the Ramanujan master theorem is appliedfor the evaluation of certain integrals by Babusci et al [22]

The technique adopted in [22] provides a very flexible andpowerful tool yielding new results encompassing differentaspects of the theory of special functions The possibility ofusing the method outlined in [22] for the integrals evaluatedin this article is a further research problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to thereviewer(s) for helpful suggestions and comments towardsthe improvement of this paper

References

[1] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985

[2] M A Chaudhry and S M Zubair ldquoGeneralized incompletegamma functions with applicationsrdquo Journal of Computationaland Applied Mathematics vol 55 no 1 pp 99ndash124 1994

[3] M A Chaudhry A Qadir M Rafique and S M ZubairldquoExtension of Eulerrsquos beta functionrdquo Journal of Computationaland Applied Mathematics vol 78 no 1 pp 19ndash32 1997

12 Journal of Mathematics

[4] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions John Wiley amp Sons New York NY USA 1984

[5] M E H Ismail and J Pitman ldquoAlgebraic evaluations of someEuler integrals duplication formulae for Appellrsquos hypergeomet-ric function 119865

1 and Brownian variationsrdquo Canadian Journal of

Mathematics vol 52 no 5 pp 961ndash981 2000[6] J Pitman ldquoBrownian motion bridge excursion and meander

characterized by sampling at independent uniform timesrdquoElectronic Journal of Probability vol 11 no 4 pp 1ndash33 1999

[7] Y A Brychkov Handbook of Special Functions DerivativesIntegrals Series and Other Formulas CRC Press New York NYUSA 2008

[8] G Dattoli S Lorenzutta G Maino A Torre and C CesaranoldquoGeneralized Hermite polynomials and super-Gaussian formsrdquoJournal of Mathematical Analysis and Applications vol 203 no3 pp 597ndash609 1996

[9] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[10] P Appell and J K de Feriet Fonctions Hypergeometriques etHyperspheriques Polynomes drsquoHermite Gauthier-Villars ParisFarnce 1926

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962

[12] M Lahiri ldquoOn a generalisation of Hermite polynomialsrdquoProceedings of the American Mathematical Society vol 27 pp117ndash121 1971

[13] G Dattoli and A Torre ldquoOperatorial methods and two variableLaguerre polynomialsrdquo Atti della Accademia delle Scienze diTorino Classe di Scienze Fisiche Matematiche e Naturali vol132 pp 1ndash7 1998

[14] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009

[15] S RomanTheUmbral Calculus Academic Press NewYorkNYUSA 1984

[16] S Khan M W Al-Saad and R Khan ldquoLaguerre-based Appellpolynomials properties and applicationsrdquo Mathematical andComputer Modelling vol 52 no 1-2 pp 247ndash259 2010

[17] S Khan M W Al-Saad and G Yasmin ldquoSome properties ofHermite-based Sheffer polynomialsrdquo Applied Mathematics andComputation vol 217 no 5 pp 2169ndash2183 2010

[18] E D Rainville Special Functions Macmillan New York NYUSA 1960

[19] E D Rainville SpecialFunctions Chelsea Publishing NewYorkNY USA 1971

[20] B C Berndt Ramanujanrsquos NotebooksmdashPart I Springer NewYork NY USA 1985

[21] G H Hardy Ramanujan Twelve Lectures on Subjects Suggestedby His Life and Work Cambridge University Press CambridgeUK 1940

[22] D Babusci G Dattoli G H E Duchamp K Gorska and K APenson ldquoDefinite integrals and operational methodsrdquo AppliedMathematics and Computation vol 219 no 6 pp 3017ndash30212012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 3

2 Euler Type Integrals

We derive the evaluations of certain integrals of the followingtype

119868119889120572120573120574119886119890

[120601 (119905) 120595 (119905)]

=1

119861 (120572 120573)int

119890

119886

(119905 minus 119886)120572minus1

(119890 minus 119905)120573minus1

(120601 (119905))120574

times119903119865119905

[

[

(119891119903)

119889120595 (119905)

(119892119905)

]

]

119889119905

(15)

We evaluate the integrals of type (15) for particularfunctions 120601(119905) and 120595(119905) by considering the following cases

Case 1 Taking120601(119905) = (1minus1199091119905)minus1205721(1minus119909

2119905)minus1205722 120595(119905) = 1119905(1minus119905)

119886 = 0 and 119890 = 120574 = 1 in (15) and using [5 page 962 (7)]

int

1

0

119905120572minus1

(1 minus 119905)120573minus1

(1 minus 1199091119905)minus1205721

(1 minus 1199092119905)minus1205722

119889119905

= 119861 (120572 120573) 1198651[120572 120572

1 120572

2 120572 + 120573 119909

1 119909

2]

(Re (120572) Re (120573) gt 0 max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 lt 1)

(16)

where 1198651denotes the Appell function [4] in the rhs

after expanding119903119865119905[

(119891119903)

119889120595(119905)

(119892119905)

] and simplifying we find the

following integral

119868119889120572120573101

[(1 minus 1199091119905)minus1205721

(1 minus 1199092119905)minus1205722

1

119905 (1 minus 119905)]

=

infin

sum

11989811198982=0

(120572)1198981+1198982

(1205721)1198981

(1205722)1198982

1199091198981

11199091198982

2

(120572 + 120573)1198981+1198982

1198981119898

2

times119903+2119865119905+2

[[

[

(119891119903) 1

2(1 minus 120572 minus 120573 minus 119898

1minus 119898

2) 1 minus

1

2(120572 + 120573 + 119898

1+ 119898

2)

4119889

(119892119905) 1 minus (120572 + 119898

1+ 119898

2) 1 minus 120573

]]

]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 lt 1)

(17)

Case 2 Taking 120601(119905) =119901

119865119902[

(119886119901)

119888119905(119890minus119905)

(119887119902)

] 120595(119905) = 1119905(119890 minus 119905)

119886 = 0 and 120574 = 1 in (15) and using [7 page 303 (1)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1

119901119865119902

[[

[

(119886119901)

119888119905 (119890 minus 119905)

(119887119902)

]]

]

119889119905

= (119890)120572+120573minus1

119861 (120572 120573)

times119901+2

119865119902+2

[[[[[

[

(119886119901) 120572 120573

1198881198902

4

(119887119902)

120572 + 120573

2120572 + 120573 + 1

2

]]]]]

]

(119890Re (120572) Re (120573) gt 010038161003816100381610038161003816arg (4 minus 1198881198902)10038161003816100381610038161003816 lt 120587)

(18)

we find the following integral

11986811988912057212057310119890

[

[

119901119865119902

[

[

(119886119901)

119888119905 (119890 minus 119905)

(119887119902)

]

]

1

119905 (119890 minus 119905)

]

]

= (119890)120572+120573minus1

infin

sum

119898=0

(119886119901)119898

(120572)119898(120573)

119898(119888119890

2

4)

119898

(119887119902)119898

((120572 + 120573)2)119898((120572 + 120573 + 1)2)

119898119898

times119903+2119865119905+2

[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 1 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905) 1 minus 120572 minus 119898 1 minus 120573 minus 119898

]]]]

]

(119890Re (120572) Re (120573) gt 0 10038161003816100381610038161003816arg (4 minus 1198881198902)1003816100381610038161003816

1003816lt 120587)

(19)

Case 3 Taking120601(119905) =119901

119865119902[

(119886119901)

119888119905(119890minus119905)

(119887119902)

]120595(119905) = 1(1199052

(119890 minus 119905)2

)

119886 = 0 and 120574 = 1 and proceeding as in Case 2 we find thefollowing integral

11986811988912057212057310119890

[[

[

119901119865119902

[[

[

(119886119901)

119888119905 (119890 minus 119905)

(119887119902)

]]

]

1

1199052(119890 minus 119905)2

]]

]

= (119890)120572+120573minus1

infin

sum

119898=0

(119886119901)119898

(120572)119898(120573)

119898(119888119890

2

4)119898

(119887119902)119898

((120572 + 120573) 2)119898((120572 + 120573 + 1) 2)

119898119898

4 Journal of Mathematics

times119903+6119865119905+8

[[[[[[

[

(119891119903) 119895 minus 120572 minus 120573

21

2(1 minus 120572 minus 120573) minus 119898 1 minus

1

2(120572 + 120573) minus 119898

4119889

1198904

(119892119905) 119895 minus 120572 minus 120573

41 minus 120572 minus 119898

22 minus 120572 minus 119898

21 minus 120573 minus 119898

22 minus 120573 minus 119898

2

]]]]]]

]

(119895 = 1 2 3 4) (119890Re (120572) Re (120573) gt 010038161003816100381610038161003816arg (4 minus 1198881198902)10038161003816100381610038161003816 lt 120587)

(20)

Case 4 Taking 120601(119905) = K(119888radic119905(119890 minus 119905)) 120595(119905) = 1119905(119890 minus 119905) 119886 = 0and 120574 = 1 in (15) where K(119911) is the complete elliptic integralof the first kind [4 page 35 (32)] defined by

K (119911) = int

1205872

0

119889120579

radic(1 minus 1199112sin2120579)(21)

and using [7 page 265 (4)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1K (119888radic119905 (119890 minus 119905)) 119889119905

=120587

2(119890)

120572+120573minus1

119861 (120572 120573)41198653

[[[[[[

[

1

21

2 120572 120573

1198882

1198902

4

1120572 + 120573

2120572 + 120573 + 1

2

]]]]]]

]

(Re (120572) Re (120573) gt 010038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(22)

we find the following integral

11986811988912057212057310119890

[K(119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)

]

=

120587

2

(119890)120572+120573minus1

infin

sum

119898=0

(12)119898(12)

119898(120572)

119898(120573)

119898(1198882

1198902

4)

119898

(1)119898((120572 + 120573)2)

119898((120572 + 120573 + 1)2)

119898119898

times119903+2119865119905+2

[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 1 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905) 1 minus 120572 minus 119898 1 minus 120573 minus 119898

]]]]

]

(Re (120572) Re (120573) gt 0 10038161003816100381610038161003816arg (4 minus 11988821198902)1003816100381610038161003816

1003816lt 120587)

(23)

Case 5 Taking 120601(119905) = E(119888radic119905(119890 minus 119905)) 120595(119905) = 1119905(119890 minus 119905) 119886 = 0and 120574 = 1 in (15) where E(119911) is the complete elliptic integralof the second kind [4 page 35 (33)] defined by

E (119911) = int

1205872

0

radic(1 minus 1199112sin2120579)119889120579 (24)

and using [7 page 280 (3)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1E (119888radic119905 (119890 minus 119905)) 119889119905

=120587

2(119890)

120572+120573minus1

119861 (120572 120573)41198653

[[[[[[

[

minus1

21

2 120572 120573

1198882

1198902

4

1120572 + 120573

2120572 + 120573 + 1

2

]]]]]]

]

(119890Re (120572) Re (120573) gt 010038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(25)

we find the following integral

11986811988912057212057310119890

[E(119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)

]

=

120587

2

(119890)120572+120573minus1

infin

sum

119898=0

(minus12)119898(12)

119898(120572)

119898(120573)

119898(1198882

1198902

4)

119898

(1)119898((120572 + 120573)2)

119898((120572 + 120573 + 1)2)

119898119898

times119903+2119865119905+2

[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 1 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905) 1 minus 120572 minus 119898 1 minus 120573 minus 119898

]]]]

]

(119890Re (120572) Re (120573) gt 0 10038161003816100381610038161003816arg (4 minus 11988821198902)1003816100381610038161003816

1003816lt 120587)

(26)

Case 6 Taking 120601(119905) = arcsin(119888radic119905(119890 minus 119905)) 120595(119905) = 1119905(119890 minus 119905)119886 = 0 and 120574 = 2 in (15) and using [7 page 173 (151)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1arcsin2 (119888radic119905 (119890 minus 119905)) 119889119905

= 1198882

(119890)120572+120573+1

119861 (120572 + 1 120573 + 1)

times51198654

[[[[

[

1 1 1 120572 + 1 120573 + 1

1198882

1198902

43

2 2

120572 + 120573 + 2

2120572 + 120573 + 3

2

]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus110038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(27)

Journal of Mathematics 5

we find the following integral

11986811988912057212057320119890

[arcsin (119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)]

=120572120573119888

2

(119890)120572+120573+1

(120572 + 120573) (120572 + 120573 + 1)

times

infin

sum

119898=0

(1)119898(1)

119898(1)

119898(120572 + 1)

119898(120573 + 1)

119898(1198882

1198902

4)119898

(32)119898(2)

119898((120572 + 120573 + 2)2)

119898((120572 + 120573 + 3)2)

119898119898

times119903+2

119865119905+2

[[[[

[

(119891119903) minus

1

2(1 + 120572 + 120573) minus 119898 minus

1

2(120572 + 120573) minus 119898

4119889

1198902

(119892119905) minus120572 minus 119898 minus120573 minus 119898

]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus110038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(28)

Case 7 Taking 120601(119905) = exp(1198882119905(119890 minus 119905)) erf(119888radic119905(119890 minus 119905)) 120595(119905) =1119905(119890 minus 119905) 119886 = 0 and 120574 = 1 in (15) and using [7 page 185 (1)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1 exp (1198882119905 (119890 minus 119905)) erf (119888radic119905 (119890 minus 119905)) 119889119905

=2

radic120587119888(119890)

120572+120573

119861(120572 +1

2 120573 +

1

2)

times31198653

[[[[[[

[

1 120572 +1

2 120573 +

1

2

1198882

1198902

43

2120572 + 120573 + 1

2120572 + 120573 + 2

2

]]]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus1

2)

(29)

where erf(119911) = (2radic120587)Erf(119911) denotes the error function [4]we find the following integral

11986811988912057212057310119890

[exp (1198882119905 (119890 minus 119905)) erf (119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)

]

=

2

radic120587

119888(119890)120572+120573

119861 (120572 + (12) 120573 + (12))

119861 (120572 120573)

times

infin

sum

119898=0

(1)119898(120572 + (12))

119898(120573 + (12))

119898(1198882

1198902

4)

119898

(32)119898((120572 + 120573 + 1)2)

119898((120572 + 120573 + 2)2)

119898119898

times119903+2119865119905+2

[[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905)

1

2

minus 120572 minus 119898

1

2

minus 120573 minus 119898

]]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus12

)

(30)

Now we establish a theorem and apply it to obtain theevaluations of certain integrals in terms of extended betafunction

3 Integrals in Terms of ExtendedBeta Function

Consider an (119903+1)-variable generating function119866(1199091 1199092

119909119903 119905)which possesses a formal (not necessarily convergent for

119905 = 0) power series expansion in 119905 such that [4 page 80 (8)]

119866 (1199091 1199092 119909

119903 119905) =

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119905119899

(31)

where each member of the generated set 119892119899(1199091 1199092

119909119903)infin

119899=0is independent of 119905 and the coefficient set 119888

119899infin

119899=0may

contain the parameters of the set 119892119899(1199091 1199092 119909

119903)infin

119899=0but is

independent of 1199091 1199092 119909

119903and 119905

Theorem 1 Let the generating function 119866(1199091 1199092 119909

119903 119905)

defined by (31) be such that 119866(1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

remains uniformly convergent for 119906 isin (119886 119888) 120588 120590 ge 0 and120588 + 120590 gt 0 Then

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(32)

where 119861(119909 119910 119887) is the extended beta function defined by (7)

Proof Applying the definition of 119866(1199091 1199092 119909

119903 119905) given in

(31) in the lhs of (32) we get

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119905119899

times int

119888

119886

(119906 minus 119886)120582+120588119899minus1

(119888 minus 119906)120583minus120582+120590119899minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

(33)

which by using (9) yields the rhs of (32)

Corollary 2 With definition (31) and notations as inTheorem 1 one has

int

1

0

119906120582minus1

(1 minus 119906)120583minus120582minus1

119866 (1199091 1199092 119909

119903 119905119906

120588

(1 minus 119906)120590

)

times exp(minus 119887

119906 (1 minus 119906)) 119889119906

6 Journal of Mathematics

=

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(34)

Proof Taking 119886 = 0 and 119888 = 1 in (32) we get (34)

Corollary 3 With definition (31) and notations as inTheorem 1 one has

int

1

minus1

(1 + 119906)120582minus1

(1 minus 119906)120583minus120582minus1

times 119866 (1199091 1199092 119909

119903 119905(1 + 119906)

120588

(1 minus 119906)120590

)

times exp(minus 4119887

(1 minus 1199062)) 119889119906

=

infin

sum

119899=0

(2)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(35)

Proof Taking 119886 = minus1 and 119888 = 1 in (32) we get (35)

Corollary 4 With definition (31) and notations as inTheorem 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)minus120582minus1

119866(1199091 1199092 119909

119903 119905(

119888 minus 119906

119906 minus 119886)

120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=2

(119888 minus 119886) 1198902119887

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903)119870

120582minus120590119899(2119887) 119905

119899

(Re (120582) gt 0Re (119887) gt 0)

(36)

Proof Taking 120583 = 0 and 120588 = minus120590 in (32) and making use ofrelation (10) we get (36)

Now we apply Theorem 1 to derive the evaluations ofcertain Euler type integrals in terms of EBF 119861(120572 120573 119887) Weconsider the following cases

Case 1 Consider the generating function of the multivariableHermite polynomials 119867(12119903)

119899(1199091 1199092 119909

119903) [8 page 602

(2021)]

exp (1199091119905 + 119909

21199052

+ sdot sdot sdot + 119909119903119905119903

)

=

infin

sum

119899=0

119867(12119903)

119899(1199091 1199092 119909

119903)119905119899

119899

(37)

Applying Theorem 1 to generating function (37) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

119903

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(38)

Taking 119903 = 3 in (38) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

3

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(39)

where119867119899(1199091 1199092 1199093) are the 3-variable Hermite polynomials

(3VHP) [9]Again taking 119903 = 2 in (38) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

2

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(1199091 1199092)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(40)

Journal of Mathematics 7

where 119867119899(1199091 1199092) are the 2-variable Hermite-Kampe de-

Feriet polynomials (2VHKdFP) [10]Next replacing 119909

1by 119909 119909

119903by 119910 and taking 119909

2=

1199093

= sdot sdot sdot = 119909119903minus1

= 0 in (38) and using the relation119867(12119903)

119899(119909 0 0 0 119910) = 119892

119903

119899(119909 119910) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

+119910(119905(119906 minus 119886)120588

(119888 minus 119906)120590

)119903

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119892119903

119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(41)

where 119892119903119899(119909 119910) are the Gould-Hopper generalized Hermite

polynomials (GHGHP) [11]Further replacing 119909

1by ]119909 and taking 119909

2= 119909

3= sdot sdot sdot =

119909119903minus1

= 0 119909119903= minus1 in (38) and using the relation119867(12119903)

119899(]119909

0 0 0 minus1) = 119867119899119903](119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+]119909119905(119906 minus 119886)120588(119888 minus 119906)120590minus (119905(119906 minus 119886)120588(119888 minus 119906)120590)119903)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899119903] (119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(42)

where119867119899119903](119909) are the generalizedHermite polynomials [12]

Furthermore taking 119903 = ] = 2 in (42) and using therelation119867

11989922(119909) = 119867

119899(119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+2119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

minus (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)2

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(43)

where119867119899(119909) are the ordinary Hermite polynomials [1]

Case 2 Consider the generating function of the 2-variableLaguerre polynomials 119871

119899(119909 119910) [13]

1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905) =

infin

sum

119899=0

119871119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(44)

which can also be expressed as

exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119899(119909 119910)

119905119899

119899 (45)

where 1198620(119909) denotes the 0th-order Tricomi function [4]

Applying Theorem 1 to generating functions (44) and(45) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(46)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(47)

respectively

8 Journal of Mathematics

Taking 119910 = 1 in (46) and (47) and using the relation119871119899(119909 1) = 119871

119899(119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(48)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(49)

respectively where 119871119899(119909) are the ordinary Laguerre polyno-

mials [1]Again taking119910 = 0 in (46) and (47) and using the relation

119871119899(119909 0) = (minus119909)

119899

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(50)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(51)respectively

Next taking 119909 = 0 in (46) and (47) and using the relation119871119899(0 119910) = 119910

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590exp(minus 119887(119888 minus 119886)

2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) (119910119905)119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(52)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(119910119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(53)

respectively Replacing 119910 by minus119909 in (53) it reduces to (50)

Case 3 Consider the generating function of the Hermite-Appell polynomials

119867119860119899(1199091 1199092 1199093) [14 page 759 (23)]

119860 (119905) exp (1199091119905 + 119909

21199052

+ 11990931199053

) =

infin

sum

119899=0

119867119860119899(1199091 1199092 1199093)119905119899

119899

(54)

Applying Theorem 1 to generating function (54) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+

3

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119860119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(55)

Journal of Mathematics 9

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (55) and

using the relation119867119860119899(119909 0 0) = 119860

119899(119909) [14 page 760 (26)]

we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(56)

where 119860119899(119909) are the Appell polynomials [4 15]

Case 4 Consider the generating function of the Laguerre-Appell polynomials

119871119860119899(119909 119910) [16]

119860 (119905) [1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905)]

=

infin

sum

119899=0

119871119860119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(57)

which can also be expressed as follows

119860 (119905) exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119860119899(119909 119910)

119905119899

119899 (58)

Applying Theorem 1 to generating functions (57) and(58) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(59)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(60)

respectivelyTaking 119910 = 0 and replacing 119909 by minus119909 in (59) and (60) and

using the relation119871119860119899(119909 0) = 119860

119899(minus119909)119899 [16 page 9 (28)]

we get (56) and

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(61)

respectively

Case 5 Consider the generating function of the Hermite-Sheffer polynomials

119867119904119899(1199091 1199092 1199093) [17]

119860 (119905) exp (1199091119867(119905) + 119909

21198672

(119905) + 11990931198673

(119905))

=

infin

sum

119899=0

119867119904119899(1199091 1199092 1199093)119905119899

119899

(62)

Applying Theorem 1 to generating function (62) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

3

sum

119895=1

119909119895119867119895

(119905(119906 minus 119886)120588

(119888 minus 119906)120590

))119889119906

10 Journal of Mathematics

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119904119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(63)

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (63) and

using the relation119867119904119899(119909 0 0) = 119904

119899(119909) [17 page 16 (23)] we

get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119867 (119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119904119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(64)

where 119904119899(119909) are the Sheffer polynomials [15 18 19]

4 Concluding Remarks

In this paper we have derived the evaluations of certain Eulertype integralsWehave also established a theoremand appliedit to obtain evaluations of some integrals in terms of EBF119861(120572 120573 119887) We remark that these results can be extended tomultivariable case

To give an example we consider the following integralrepresentation [5 page 965 (20)] of Lauricellarsquos multiplehypergeometric series 119865(119899)

119863[4]

int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

119889119905

= 119861 (120572 120573) 119865(119899)

119863[120572 120572

1 1205722 120572

119899 120572 + 120573 119909

1 1199092 119909

119899]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(65)

Now taking 120601(119905) = prod119899

119894=1(1 minus 119909

119894119905)minus120572119894 120595(119905) = 1119905(1 minus 119905)

119886 = 0 and 119890 = 120574 = 1 in (15) using integral (65) in the rhs

after expanding119903119865119905[

(119891119903)

119889120595(119905)

(119892119905)

] in series and simplifying weget

119868119889120572120573101

[

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

1

119905 (1 minus 119905)]

=

infin

sum

11989811198982119898119899=0

(120572)1198981+1198982+sdotsdotsdot+119898

119899

(1205721)1198981

(1205722)1198982

sdot sdot sdot (120572119899)119898119899

(120572 + 120573)1198981+1198982+sdotsdotsdot+119898

119899

1199091198981

1

1198981

1199091198982

2

1198982sdot sdot sdot

119909119898119899

119899

119898119899

times119903+2119865119905+2

[[[[[[[

[

(119891119903) 1

2(1 minus 120572 minus 120573 minus

119899

sum

119894=1

119898119894) 1 minus

1

2(120572 + 120573 +

119899

sum

119894=1

119898119894)

4119889

(119892119905) 1 minus (120572 +

119899

sum

119894=1

119898119894) 1 minus 120573

]]]]]]]

]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(66)

We note that for 1205723= 120572

4= sdot sdot sdot = 0 (66) reduces to (17)

Further we extendTheorem 1 as follows

Theorem 5 Let the conditions for 119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

11989911989811198982119898119904=0

(119888 minus 119886)120583+(120588+120590)119899+119898

1+1198982+sdotsdotsdot+119898

119904minus1

times 119888119899119892119899(1199091 1199092 119909

119903)

times 119861(120582 + 120588119899 +

119904

sum

119894=1

119898119894 120583 minus 120582 + 120590119899 119887)

119904

prod

119894=1

(120572119894)119898119894

119910119898119894

119894

119898119894

119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(67)

Journal of Mathematics 11

The proof of this theorem is similar to that of Theorem 1The following are the consequences of Theorem 5

Corollary 6 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and119904 = 2 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

2

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899)

times 1198651[120582 + 120588119899 120572

1 1205722 120583 + (120588 + 120590) 119899

(119888 minus 119886) 1199101 (119888 minus 119886) 119910

2] 119905119899

(Re (120583) gt Re (120582) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(68)

Corollary 7 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and120588 = 120590 = 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905 (119906 minus 119886) (119888 minus 119906)) 119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904] 119905119899

(Re (120583) gt Re (120582) gt 0)

(69)

Example 8 Applying Corollary 7 to generating function (37)of the multivariable Hermite polynomials 119867(12119903)

119899(1199091 1199092

119909119903) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times exp(119903

sum

119895=1

119909119895(119905 (119906 minus 119886) (119888 minus 119906))

119895

)119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904]119905119899

119899

(Re (120583) gt Re (120582) gt 0)

(70)Remark 9 Taking 119886 = 0 and 119888 = 1 in (67) we get an extensionof integral (34)

Remark 10 Taking 119886 = minus1 and 119888 = 1 in (67) we get anextension of integral (35)

Clearly it appears that by applying Theorem 5 andCorollaries 6 and 7 to the generating functions defined by(37) (44) (45) (54) (57) (58) and (62) and also to someother generating functions a number of interesting integralscan be obtained

Further it is remarked that the operational methodsprovide useful tools to estimate specific theorems in variousfields of analysisThis approach requires the validation of anyof its consequences by a more rigorous procedure This isindeed the case of the Ramanujan master theorem [20 21]An operational method already employed to formulate ageneralization of the Ramanujan master theorem is appliedfor the evaluation of certain integrals by Babusci et al [22]

The technique adopted in [22] provides a very flexible andpowerful tool yielding new results encompassing differentaspects of the theory of special functions The possibility ofusing the method outlined in [22] for the integrals evaluatedin this article is a further research problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to thereviewer(s) for helpful suggestions and comments towardsthe improvement of this paper

References

[1] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985

[2] M A Chaudhry and S M Zubair ldquoGeneralized incompletegamma functions with applicationsrdquo Journal of Computationaland Applied Mathematics vol 55 no 1 pp 99ndash124 1994

[3] M A Chaudhry A Qadir M Rafique and S M ZubairldquoExtension of Eulerrsquos beta functionrdquo Journal of Computationaland Applied Mathematics vol 78 no 1 pp 19ndash32 1997

12 Journal of Mathematics

[4] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions John Wiley amp Sons New York NY USA 1984

[5] M E H Ismail and J Pitman ldquoAlgebraic evaluations of someEuler integrals duplication formulae for Appellrsquos hypergeomet-ric function 119865

1 and Brownian variationsrdquo Canadian Journal of

Mathematics vol 52 no 5 pp 961ndash981 2000[6] J Pitman ldquoBrownian motion bridge excursion and meander

characterized by sampling at independent uniform timesrdquoElectronic Journal of Probability vol 11 no 4 pp 1ndash33 1999

[7] Y A Brychkov Handbook of Special Functions DerivativesIntegrals Series and Other Formulas CRC Press New York NYUSA 2008

[8] G Dattoli S Lorenzutta G Maino A Torre and C CesaranoldquoGeneralized Hermite polynomials and super-Gaussian formsrdquoJournal of Mathematical Analysis and Applications vol 203 no3 pp 597ndash609 1996

[9] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[10] P Appell and J K de Feriet Fonctions Hypergeometriques etHyperspheriques Polynomes drsquoHermite Gauthier-Villars ParisFarnce 1926

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962

[12] M Lahiri ldquoOn a generalisation of Hermite polynomialsrdquoProceedings of the American Mathematical Society vol 27 pp117ndash121 1971

[13] G Dattoli and A Torre ldquoOperatorial methods and two variableLaguerre polynomialsrdquo Atti della Accademia delle Scienze diTorino Classe di Scienze Fisiche Matematiche e Naturali vol132 pp 1ndash7 1998

[14] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009

[15] S RomanTheUmbral Calculus Academic Press NewYorkNYUSA 1984

[16] S Khan M W Al-Saad and R Khan ldquoLaguerre-based Appellpolynomials properties and applicationsrdquo Mathematical andComputer Modelling vol 52 no 1-2 pp 247ndash259 2010

[17] S Khan M W Al-Saad and G Yasmin ldquoSome properties ofHermite-based Sheffer polynomialsrdquo Applied Mathematics andComputation vol 217 no 5 pp 2169ndash2183 2010

[18] E D Rainville Special Functions Macmillan New York NYUSA 1960

[19] E D Rainville SpecialFunctions Chelsea Publishing NewYorkNY USA 1971

[20] B C Berndt Ramanujanrsquos NotebooksmdashPart I Springer NewYork NY USA 1985

[21] G H Hardy Ramanujan Twelve Lectures on Subjects Suggestedby His Life and Work Cambridge University Press CambridgeUK 1940

[22] D Babusci G Dattoli G H E Duchamp K Gorska and K APenson ldquoDefinite integrals and operational methodsrdquo AppliedMathematics and Computation vol 219 no 6 pp 3017ndash30212012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Mathematics

times119903+6119865119905+8

[[[[[[

[

(119891119903) 119895 minus 120572 minus 120573

21

2(1 minus 120572 minus 120573) minus 119898 1 minus

1

2(120572 + 120573) minus 119898

4119889

1198904

(119892119905) 119895 minus 120572 minus 120573

41 minus 120572 minus 119898

22 minus 120572 minus 119898

21 minus 120573 minus 119898

22 minus 120573 minus 119898

2

]]]]]]

]

(119895 = 1 2 3 4) (119890Re (120572) Re (120573) gt 010038161003816100381610038161003816arg (4 minus 1198881198902)10038161003816100381610038161003816 lt 120587)

(20)

Case 4 Taking 120601(119905) = K(119888radic119905(119890 minus 119905)) 120595(119905) = 1119905(119890 minus 119905) 119886 = 0and 120574 = 1 in (15) where K(119911) is the complete elliptic integralof the first kind [4 page 35 (32)] defined by

K (119911) = int

1205872

0

119889120579

radic(1 minus 1199112sin2120579)(21)

and using [7 page 265 (4)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1K (119888radic119905 (119890 minus 119905)) 119889119905

=120587

2(119890)

120572+120573minus1

119861 (120572 120573)41198653

[[[[[[

[

1

21

2 120572 120573

1198882

1198902

4

1120572 + 120573

2120572 + 120573 + 1

2

]]]]]]

]

(Re (120572) Re (120573) gt 010038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(22)

we find the following integral

11986811988912057212057310119890

[K(119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)

]

=

120587

2

(119890)120572+120573minus1

infin

sum

119898=0

(12)119898(12)

119898(120572)

119898(120573)

119898(1198882

1198902

4)

119898

(1)119898((120572 + 120573)2)

119898((120572 + 120573 + 1)2)

119898119898

times119903+2119865119905+2

[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 1 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905) 1 minus 120572 minus 119898 1 minus 120573 minus 119898

]]]]

]

(Re (120572) Re (120573) gt 0 10038161003816100381610038161003816arg (4 minus 11988821198902)1003816100381610038161003816

1003816lt 120587)

(23)

Case 5 Taking 120601(119905) = E(119888radic119905(119890 minus 119905)) 120595(119905) = 1119905(119890 minus 119905) 119886 = 0and 120574 = 1 in (15) where E(119911) is the complete elliptic integralof the second kind [4 page 35 (33)] defined by

E (119911) = int

1205872

0

radic(1 minus 1199112sin2120579)119889120579 (24)

and using [7 page 280 (3)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1E (119888radic119905 (119890 minus 119905)) 119889119905

=120587

2(119890)

120572+120573minus1

119861 (120572 120573)41198653

[[[[[[

[

minus1

21

2 120572 120573

1198882

1198902

4

1120572 + 120573

2120572 + 120573 + 1

2

]]]]]]

]

(119890Re (120572) Re (120573) gt 010038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(25)

we find the following integral

11986811988912057212057310119890

[E(119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)

]

=

120587

2

(119890)120572+120573minus1

infin

sum

119898=0

(minus12)119898(12)

119898(120572)

119898(120573)

119898(1198882

1198902

4)

119898

(1)119898((120572 + 120573)2)

119898((120572 + 120573 + 1)2)

119898119898

times119903+2119865119905+2

[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 1 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905) 1 minus 120572 minus 119898 1 minus 120573 minus 119898

]]]]

]

(119890Re (120572) Re (120573) gt 0 10038161003816100381610038161003816arg (4 minus 11988821198902)1003816100381610038161003816

1003816lt 120587)

(26)

Case 6 Taking 120601(119905) = arcsin(119888radic119905(119890 minus 119905)) 120595(119905) = 1119905(119890 minus 119905)119886 = 0 and 120574 = 2 in (15) and using [7 page 173 (151)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1arcsin2 (119888radic119905 (119890 minus 119905)) 119889119905

= 1198882

(119890)120572+120573+1

119861 (120572 + 1 120573 + 1)

times51198654

[[[[

[

1 1 1 120572 + 1 120573 + 1

1198882

1198902

43

2 2

120572 + 120573 + 2

2120572 + 120573 + 3

2

]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus110038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(27)

Journal of Mathematics 5

we find the following integral

11986811988912057212057320119890

[arcsin (119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)]

=120572120573119888

2

(119890)120572+120573+1

(120572 + 120573) (120572 + 120573 + 1)

times

infin

sum

119898=0

(1)119898(1)

119898(1)

119898(120572 + 1)

119898(120573 + 1)

119898(1198882

1198902

4)119898

(32)119898(2)

119898((120572 + 120573 + 2)2)

119898((120572 + 120573 + 3)2)

119898119898

times119903+2

119865119905+2

[[[[

[

(119891119903) minus

1

2(1 + 120572 + 120573) minus 119898 minus

1

2(120572 + 120573) minus 119898

4119889

1198902

(119892119905) minus120572 minus 119898 minus120573 minus 119898

]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus110038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(28)

Case 7 Taking 120601(119905) = exp(1198882119905(119890 minus 119905)) erf(119888radic119905(119890 minus 119905)) 120595(119905) =1119905(119890 minus 119905) 119886 = 0 and 120574 = 1 in (15) and using [7 page 185 (1)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1 exp (1198882119905 (119890 minus 119905)) erf (119888radic119905 (119890 minus 119905)) 119889119905

=2

radic120587119888(119890)

120572+120573

119861(120572 +1

2 120573 +

1

2)

times31198653

[[[[[[

[

1 120572 +1

2 120573 +

1

2

1198882

1198902

43

2120572 + 120573 + 1

2120572 + 120573 + 2

2

]]]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus1

2)

(29)

where erf(119911) = (2radic120587)Erf(119911) denotes the error function [4]we find the following integral

11986811988912057212057310119890

[exp (1198882119905 (119890 minus 119905)) erf (119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)

]

=

2

radic120587

119888(119890)120572+120573

119861 (120572 + (12) 120573 + (12))

119861 (120572 120573)

times

infin

sum

119898=0

(1)119898(120572 + (12))

119898(120573 + (12))

119898(1198882

1198902

4)

119898

(32)119898((120572 + 120573 + 1)2)

119898((120572 + 120573 + 2)2)

119898119898

times119903+2119865119905+2

[[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905)

1

2

minus 120572 minus 119898

1

2

minus 120573 minus 119898

]]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus12

)

(30)

Now we establish a theorem and apply it to obtain theevaluations of certain integrals in terms of extended betafunction

3 Integrals in Terms of ExtendedBeta Function

Consider an (119903+1)-variable generating function119866(1199091 1199092

119909119903 119905)which possesses a formal (not necessarily convergent for

119905 = 0) power series expansion in 119905 such that [4 page 80 (8)]

119866 (1199091 1199092 119909

119903 119905) =

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119905119899

(31)

where each member of the generated set 119892119899(1199091 1199092

119909119903)infin

119899=0is independent of 119905 and the coefficient set 119888

119899infin

119899=0may

contain the parameters of the set 119892119899(1199091 1199092 119909

119903)infin

119899=0but is

independent of 1199091 1199092 119909

119903and 119905

Theorem 1 Let the generating function 119866(1199091 1199092 119909

119903 119905)

defined by (31) be such that 119866(1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

remains uniformly convergent for 119906 isin (119886 119888) 120588 120590 ge 0 and120588 + 120590 gt 0 Then

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(32)

where 119861(119909 119910 119887) is the extended beta function defined by (7)

Proof Applying the definition of 119866(1199091 1199092 119909

119903 119905) given in

(31) in the lhs of (32) we get

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119905119899

times int

119888

119886

(119906 minus 119886)120582+120588119899minus1

(119888 minus 119906)120583minus120582+120590119899minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

(33)

which by using (9) yields the rhs of (32)

Corollary 2 With definition (31) and notations as inTheorem 1 one has

int

1

0

119906120582minus1

(1 minus 119906)120583minus120582minus1

119866 (1199091 1199092 119909

119903 119905119906

120588

(1 minus 119906)120590

)

times exp(minus 119887

119906 (1 minus 119906)) 119889119906

6 Journal of Mathematics

=

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(34)

Proof Taking 119886 = 0 and 119888 = 1 in (32) we get (34)

Corollary 3 With definition (31) and notations as inTheorem 1 one has

int

1

minus1

(1 + 119906)120582minus1

(1 minus 119906)120583minus120582minus1

times 119866 (1199091 1199092 119909

119903 119905(1 + 119906)

120588

(1 minus 119906)120590

)

times exp(minus 4119887

(1 minus 1199062)) 119889119906

=

infin

sum

119899=0

(2)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(35)

Proof Taking 119886 = minus1 and 119888 = 1 in (32) we get (35)

Corollary 4 With definition (31) and notations as inTheorem 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)minus120582minus1

119866(1199091 1199092 119909

119903 119905(

119888 minus 119906

119906 minus 119886)

120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=2

(119888 minus 119886) 1198902119887

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903)119870

120582minus120590119899(2119887) 119905

119899

(Re (120582) gt 0Re (119887) gt 0)

(36)

Proof Taking 120583 = 0 and 120588 = minus120590 in (32) and making use ofrelation (10) we get (36)

Now we apply Theorem 1 to derive the evaluations ofcertain Euler type integrals in terms of EBF 119861(120572 120573 119887) Weconsider the following cases

Case 1 Consider the generating function of the multivariableHermite polynomials 119867(12119903)

119899(1199091 1199092 119909

119903) [8 page 602

(2021)]

exp (1199091119905 + 119909

21199052

+ sdot sdot sdot + 119909119903119905119903

)

=

infin

sum

119899=0

119867(12119903)

119899(1199091 1199092 119909

119903)119905119899

119899

(37)

Applying Theorem 1 to generating function (37) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

119903

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(38)

Taking 119903 = 3 in (38) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

3

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(39)

where119867119899(1199091 1199092 1199093) are the 3-variable Hermite polynomials

(3VHP) [9]Again taking 119903 = 2 in (38) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

2

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(1199091 1199092)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(40)

Journal of Mathematics 7

where 119867119899(1199091 1199092) are the 2-variable Hermite-Kampe de-

Feriet polynomials (2VHKdFP) [10]Next replacing 119909

1by 119909 119909

119903by 119910 and taking 119909

2=

1199093

= sdot sdot sdot = 119909119903minus1

= 0 in (38) and using the relation119867(12119903)

119899(119909 0 0 0 119910) = 119892

119903

119899(119909 119910) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

+119910(119905(119906 minus 119886)120588

(119888 minus 119906)120590

)119903

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119892119903

119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(41)

where 119892119903119899(119909 119910) are the Gould-Hopper generalized Hermite

polynomials (GHGHP) [11]Further replacing 119909

1by ]119909 and taking 119909

2= 119909

3= sdot sdot sdot =

119909119903minus1

= 0 119909119903= minus1 in (38) and using the relation119867(12119903)

119899(]119909

0 0 0 minus1) = 119867119899119903](119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+]119909119905(119906 minus 119886)120588(119888 minus 119906)120590minus (119905(119906 minus 119886)120588(119888 minus 119906)120590)119903)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899119903] (119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(42)

where119867119899119903](119909) are the generalizedHermite polynomials [12]

Furthermore taking 119903 = ] = 2 in (42) and using therelation119867

11989922(119909) = 119867

119899(119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+2119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

minus (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)2

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(43)

where119867119899(119909) are the ordinary Hermite polynomials [1]

Case 2 Consider the generating function of the 2-variableLaguerre polynomials 119871

119899(119909 119910) [13]

1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905) =

infin

sum

119899=0

119871119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(44)

which can also be expressed as

exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119899(119909 119910)

119905119899

119899 (45)

where 1198620(119909) denotes the 0th-order Tricomi function [4]

Applying Theorem 1 to generating functions (44) and(45) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(46)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(47)

respectively

8 Journal of Mathematics

Taking 119910 = 1 in (46) and (47) and using the relation119871119899(119909 1) = 119871

119899(119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(48)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(49)

respectively where 119871119899(119909) are the ordinary Laguerre polyno-

mials [1]Again taking119910 = 0 in (46) and (47) and using the relation

119871119899(119909 0) = (minus119909)

119899

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(50)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(51)respectively

Next taking 119909 = 0 in (46) and (47) and using the relation119871119899(0 119910) = 119910

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590exp(minus 119887(119888 minus 119886)

2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) (119910119905)119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(52)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(119910119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(53)

respectively Replacing 119910 by minus119909 in (53) it reduces to (50)

Case 3 Consider the generating function of the Hermite-Appell polynomials

119867119860119899(1199091 1199092 1199093) [14 page 759 (23)]

119860 (119905) exp (1199091119905 + 119909

21199052

+ 11990931199053

) =

infin

sum

119899=0

119867119860119899(1199091 1199092 1199093)119905119899

119899

(54)

Applying Theorem 1 to generating function (54) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+

3

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119860119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(55)

Journal of Mathematics 9

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (55) and

using the relation119867119860119899(119909 0 0) = 119860

119899(119909) [14 page 760 (26)]

we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(56)

where 119860119899(119909) are the Appell polynomials [4 15]

Case 4 Consider the generating function of the Laguerre-Appell polynomials

119871119860119899(119909 119910) [16]

119860 (119905) [1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905)]

=

infin

sum

119899=0

119871119860119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(57)

which can also be expressed as follows

119860 (119905) exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119860119899(119909 119910)

119905119899

119899 (58)

Applying Theorem 1 to generating functions (57) and(58) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(59)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(60)

respectivelyTaking 119910 = 0 and replacing 119909 by minus119909 in (59) and (60) and

using the relation119871119860119899(119909 0) = 119860

119899(minus119909)119899 [16 page 9 (28)]

we get (56) and

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(61)

respectively

Case 5 Consider the generating function of the Hermite-Sheffer polynomials

119867119904119899(1199091 1199092 1199093) [17]

119860 (119905) exp (1199091119867(119905) + 119909

21198672

(119905) + 11990931198673

(119905))

=

infin

sum

119899=0

119867119904119899(1199091 1199092 1199093)119905119899

119899

(62)

Applying Theorem 1 to generating function (62) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

3

sum

119895=1

119909119895119867119895

(119905(119906 minus 119886)120588

(119888 minus 119906)120590

))119889119906

10 Journal of Mathematics

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119904119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(63)

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (63) and

using the relation119867119904119899(119909 0 0) = 119904

119899(119909) [17 page 16 (23)] we

get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119867 (119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119904119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(64)

where 119904119899(119909) are the Sheffer polynomials [15 18 19]

4 Concluding Remarks

In this paper we have derived the evaluations of certain Eulertype integralsWehave also established a theoremand appliedit to obtain evaluations of some integrals in terms of EBF119861(120572 120573 119887) We remark that these results can be extended tomultivariable case

To give an example we consider the following integralrepresentation [5 page 965 (20)] of Lauricellarsquos multiplehypergeometric series 119865(119899)

119863[4]

int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

119889119905

= 119861 (120572 120573) 119865(119899)

119863[120572 120572

1 1205722 120572

119899 120572 + 120573 119909

1 1199092 119909

119899]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(65)

Now taking 120601(119905) = prod119899

119894=1(1 minus 119909

119894119905)minus120572119894 120595(119905) = 1119905(1 minus 119905)

119886 = 0 and 119890 = 120574 = 1 in (15) using integral (65) in the rhs

after expanding119903119865119905[

(119891119903)

119889120595(119905)

(119892119905)

] in series and simplifying weget

119868119889120572120573101

[

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

1

119905 (1 minus 119905)]

=

infin

sum

11989811198982119898119899=0

(120572)1198981+1198982+sdotsdotsdot+119898

119899

(1205721)1198981

(1205722)1198982

sdot sdot sdot (120572119899)119898119899

(120572 + 120573)1198981+1198982+sdotsdotsdot+119898

119899

1199091198981

1

1198981

1199091198982

2

1198982sdot sdot sdot

119909119898119899

119899

119898119899

times119903+2119865119905+2

[[[[[[[

[

(119891119903) 1

2(1 minus 120572 minus 120573 minus

119899

sum

119894=1

119898119894) 1 minus

1

2(120572 + 120573 +

119899

sum

119894=1

119898119894)

4119889

(119892119905) 1 minus (120572 +

119899

sum

119894=1

119898119894) 1 minus 120573

]]]]]]]

]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(66)

We note that for 1205723= 120572

4= sdot sdot sdot = 0 (66) reduces to (17)

Further we extendTheorem 1 as follows

Theorem 5 Let the conditions for 119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

11989911989811198982119898119904=0

(119888 minus 119886)120583+(120588+120590)119899+119898

1+1198982+sdotsdotsdot+119898

119904minus1

times 119888119899119892119899(1199091 1199092 119909

119903)

times 119861(120582 + 120588119899 +

119904

sum

119894=1

119898119894 120583 minus 120582 + 120590119899 119887)

119904

prod

119894=1

(120572119894)119898119894

119910119898119894

119894

119898119894

119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(67)

Journal of Mathematics 11

The proof of this theorem is similar to that of Theorem 1The following are the consequences of Theorem 5

Corollary 6 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and119904 = 2 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

2

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899)

times 1198651[120582 + 120588119899 120572

1 1205722 120583 + (120588 + 120590) 119899

(119888 minus 119886) 1199101 (119888 minus 119886) 119910

2] 119905119899

(Re (120583) gt Re (120582) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(68)

Corollary 7 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and120588 = 120590 = 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905 (119906 minus 119886) (119888 minus 119906)) 119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904] 119905119899

(Re (120583) gt Re (120582) gt 0)

(69)

Example 8 Applying Corollary 7 to generating function (37)of the multivariable Hermite polynomials 119867(12119903)

119899(1199091 1199092

119909119903) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times exp(119903

sum

119895=1

119909119895(119905 (119906 minus 119886) (119888 minus 119906))

119895

)119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904]119905119899

119899

(Re (120583) gt Re (120582) gt 0)

(70)Remark 9 Taking 119886 = 0 and 119888 = 1 in (67) we get an extensionof integral (34)

Remark 10 Taking 119886 = minus1 and 119888 = 1 in (67) we get anextension of integral (35)

Clearly it appears that by applying Theorem 5 andCorollaries 6 and 7 to the generating functions defined by(37) (44) (45) (54) (57) (58) and (62) and also to someother generating functions a number of interesting integralscan be obtained

Further it is remarked that the operational methodsprovide useful tools to estimate specific theorems in variousfields of analysisThis approach requires the validation of anyof its consequences by a more rigorous procedure This isindeed the case of the Ramanujan master theorem [20 21]An operational method already employed to formulate ageneralization of the Ramanujan master theorem is appliedfor the evaluation of certain integrals by Babusci et al [22]

The technique adopted in [22] provides a very flexible andpowerful tool yielding new results encompassing differentaspects of the theory of special functions The possibility ofusing the method outlined in [22] for the integrals evaluatedin this article is a further research problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to thereviewer(s) for helpful suggestions and comments towardsthe improvement of this paper

References

[1] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985

[2] M A Chaudhry and S M Zubair ldquoGeneralized incompletegamma functions with applicationsrdquo Journal of Computationaland Applied Mathematics vol 55 no 1 pp 99ndash124 1994

[3] M A Chaudhry A Qadir M Rafique and S M ZubairldquoExtension of Eulerrsquos beta functionrdquo Journal of Computationaland Applied Mathematics vol 78 no 1 pp 19ndash32 1997

12 Journal of Mathematics

[4] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions John Wiley amp Sons New York NY USA 1984

[5] M E H Ismail and J Pitman ldquoAlgebraic evaluations of someEuler integrals duplication formulae for Appellrsquos hypergeomet-ric function 119865

1 and Brownian variationsrdquo Canadian Journal of

Mathematics vol 52 no 5 pp 961ndash981 2000[6] J Pitman ldquoBrownian motion bridge excursion and meander

characterized by sampling at independent uniform timesrdquoElectronic Journal of Probability vol 11 no 4 pp 1ndash33 1999

[7] Y A Brychkov Handbook of Special Functions DerivativesIntegrals Series and Other Formulas CRC Press New York NYUSA 2008

[8] G Dattoli S Lorenzutta G Maino A Torre and C CesaranoldquoGeneralized Hermite polynomials and super-Gaussian formsrdquoJournal of Mathematical Analysis and Applications vol 203 no3 pp 597ndash609 1996

[9] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[10] P Appell and J K de Feriet Fonctions Hypergeometriques etHyperspheriques Polynomes drsquoHermite Gauthier-Villars ParisFarnce 1926

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962

[12] M Lahiri ldquoOn a generalisation of Hermite polynomialsrdquoProceedings of the American Mathematical Society vol 27 pp117ndash121 1971

[13] G Dattoli and A Torre ldquoOperatorial methods and two variableLaguerre polynomialsrdquo Atti della Accademia delle Scienze diTorino Classe di Scienze Fisiche Matematiche e Naturali vol132 pp 1ndash7 1998

[14] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009

[15] S RomanTheUmbral Calculus Academic Press NewYorkNYUSA 1984

[16] S Khan M W Al-Saad and R Khan ldquoLaguerre-based Appellpolynomials properties and applicationsrdquo Mathematical andComputer Modelling vol 52 no 1-2 pp 247ndash259 2010

[17] S Khan M W Al-Saad and G Yasmin ldquoSome properties ofHermite-based Sheffer polynomialsrdquo Applied Mathematics andComputation vol 217 no 5 pp 2169ndash2183 2010

[18] E D Rainville Special Functions Macmillan New York NYUSA 1960

[19] E D Rainville SpecialFunctions Chelsea Publishing NewYorkNY USA 1971

[20] B C Berndt Ramanujanrsquos NotebooksmdashPart I Springer NewYork NY USA 1985

[21] G H Hardy Ramanujan Twelve Lectures on Subjects Suggestedby His Life and Work Cambridge University Press CambridgeUK 1940

[22] D Babusci G Dattoli G H E Duchamp K Gorska and K APenson ldquoDefinite integrals and operational methodsrdquo AppliedMathematics and Computation vol 219 no 6 pp 3017ndash30212012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

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Stochastic AnalysisInternational Journal of

Journal of Mathematics 5

we find the following integral

11986811988912057212057320119890

[arcsin (119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)]

=120572120573119888

2

(119890)120572+120573+1

(120572 + 120573) (120572 + 120573 + 1)

times

infin

sum

119898=0

(1)119898(1)

119898(1)

119898(120572 + 1)

119898(120573 + 1)

119898(1198882

1198902

4)119898

(32)119898(2)

119898((120572 + 120573 + 2)2)

119898((120572 + 120573 + 3)2)

119898119898

times119903+2

119865119905+2

[[[[

[

(119891119903) minus

1

2(1 + 120572 + 120573) minus 119898 minus

1

2(120572 + 120573) minus 119898

4119889

1198902

(119892119905) minus120572 minus 119898 minus120573 minus 119898

]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus110038161003816100381610038161003816arg (4 minus 11988821198902)10038161003816100381610038161003816 lt 120587)

(28)

Case 7 Taking 120601(119905) = exp(1198882119905(119890 minus 119905)) erf(119888radic119905(119890 minus 119905)) 120595(119905) =1119905(119890 minus 119905) 119886 = 0 and 120574 = 1 in (15) and using [7 page 185 (1)]

int

119890

0

119905120572minus1

(119890 minus 119905)120573minus1 exp (1198882119905 (119890 minus 119905)) erf (119888radic119905 (119890 minus 119905)) 119889119905

=2

radic120587119888(119890)

120572+120573

119861(120572 +1

2 120573 +

1

2)

times31198653

[[[[[[

[

1 120572 +1

2 120573 +

1

2

1198882

1198902

43

2120572 + 120573 + 1

2120572 + 120573 + 2

2

]]]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus1

2)

(29)

where erf(119911) = (2radic120587)Erf(119911) denotes the error function [4]we find the following integral

11986811988912057212057310119890

[exp (1198882119905 (119890 minus 119905)) erf (119888radic119905 (119890 minus 119905)) 1

119905 (119890 minus 119905)

]

=

2

radic120587

119888(119890)120572+120573

119861 (120572 + (12) 120573 + (12))

119861 (120572 120573)

times

infin

sum

119898=0

(1)119898(120572 + (12))

119898(120573 + (12))

119898(1198882

1198902

4)

119898

(32)119898((120572 + 120573 + 1)2)

119898((120572 + 120573 + 2)2)

119898119898

times119903+2119865119905+2

[[[[[

[

(119891119903)

1

2

(1 minus 120572 minus 120573) minus 119898 minus

1

2

(120572 + 120573) minus 119898

4119889

1198902

(119892119905)

1

2

minus 120572 minus 119898

1

2

minus 120573 minus 119898

]]]]]

]

(119890 gt 0Re (120572) Re (120573) gt minus12

)

(30)

Now we establish a theorem and apply it to obtain theevaluations of certain integrals in terms of extended betafunction

3 Integrals in Terms of ExtendedBeta Function

Consider an (119903+1)-variable generating function119866(1199091 1199092

119909119903 119905)which possesses a formal (not necessarily convergent for

119905 = 0) power series expansion in 119905 such that [4 page 80 (8)]

119866 (1199091 1199092 119909

119903 119905) =

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119905119899

(31)

where each member of the generated set 119892119899(1199091 1199092

119909119903)infin

119899=0is independent of 119905 and the coefficient set 119888

119899infin

119899=0may

contain the parameters of the set 119892119899(1199091 1199092 119909

119903)infin

119899=0but is

independent of 1199091 1199092 119909

119903and 119905

Theorem 1 Let the generating function 119866(1199091 1199092 119909

119903 119905)

defined by (31) be such that 119866(1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

remains uniformly convergent for 119906 isin (119886 119888) 120588 120590 ge 0 and120588 + 120590 gt 0 Then

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(32)

where 119861(119909 119910 119887) is the extended beta function defined by (7)

Proof Applying the definition of 119866(1199091 1199092 119909

119903 119905) given in

(31) in the lhs of (32) we get

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119905119899

times int

119888

119886

(119906 minus 119886)120582+120588119899minus1

(119888 minus 119906)120583minus120582+120590119899minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

(33)

which by using (9) yields the rhs of (32)

Corollary 2 With definition (31) and notations as inTheorem 1 one has

int

1

0

119906120582minus1

(1 minus 119906)120583minus120582minus1

119866 (1199091 1199092 119909

119903 119905119906

120588

(1 minus 119906)120590

)

times exp(minus 119887

119906 (1 minus 119906)) 119889119906

6 Journal of Mathematics

=

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(34)

Proof Taking 119886 = 0 and 119888 = 1 in (32) we get (34)

Corollary 3 With definition (31) and notations as inTheorem 1 one has

int

1

minus1

(1 + 119906)120582minus1

(1 minus 119906)120583minus120582minus1

times 119866 (1199091 1199092 119909

119903 119905(1 + 119906)

120588

(1 minus 119906)120590

)

times exp(minus 4119887

(1 minus 1199062)) 119889119906

=

infin

sum

119899=0

(2)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(35)

Proof Taking 119886 = minus1 and 119888 = 1 in (32) we get (35)

Corollary 4 With definition (31) and notations as inTheorem 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)minus120582minus1

119866(1199091 1199092 119909

119903 119905(

119888 minus 119906

119906 minus 119886)

120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=2

(119888 minus 119886) 1198902119887

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903)119870

120582minus120590119899(2119887) 119905

119899

(Re (120582) gt 0Re (119887) gt 0)

(36)

Proof Taking 120583 = 0 and 120588 = minus120590 in (32) and making use ofrelation (10) we get (36)

Now we apply Theorem 1 to derive the evaluations ofcertain Euler type integrals in terms of EBF 119861(120572 120573 119887) Weconsider the following cases

Case 1 Consider the generating function of the multivariableHermite polynomials 119867(12119903)

119899(1199091 1199092 119909

119903) [8 page 602

(2021)]

exp (1199091119905 + 119909

21199052

+ sdot sdot sdot + 119909119903119905119903

)

=

infin

sum

119899=0

119867(12119903)

119899(1199091 1199092 119909

119903)119905119899

119899

(37)

Applying Theorem 1 to generating function (37) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

119903

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(38)

Taking 119903 = 3 in (38) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

3

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(39)

where119867119899(1199091 1199092 1199093) are the 3-variable Hermite polynomials

(3VHP) [9]Again taking 119903 = 2 in (38) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

2

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(1199091 1199092)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(40)

Journal of Mathematics 7

where 119867119899(1199091 1199092) are the 2-variable Hermite-Kampe de-

Feriet polynomials (2VHKdFP) [10]Next replacing 119909

1by 119909 119909

119903by 119910 and taking 119909

2=

1199093

= sdot sdot sdot = 119909119903minus1

= 0 in (38) and using the relation119867(12119903)

119899(119909 0 0 0 119910) = 119892

119903

119899(119909 119910) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

+119910(119905(119906 minus 119886)120588

(119888 minus 119906)120590

)119903

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119892119903

119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(41)

where 119892119903119899(119909 119910) are the Gould-Hopper generalized Hermite

polynomials (GHGHP) [11]Further replacing 119909

1by ]119909 and taking 119909

2= 119909

3= sdot sdot sdot =

119909119903minus1

= 0 119909119903= minus1 in (38) and using the relation119867(12119903)

119899(]119909

0 0 0 minus1) = 119867119899119903](119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+]119909119905(119906 minus 119886)120588(119888 minus 119906)120590minus (119905(119906 minus 119886)120588(119888 minus 119906)120590)119903)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899119903] (119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(42)

where119867119899119903](119909) are the generalizedHermite polynomials [12]

Furthermore taking 119903 = ] = 2 in (42) and using therelation119867

11989922(119909) = 119867

119899(119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+2119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

minus (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)2

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(43)

where119867119899(119909) are the ordinary Hermite polynomials [1]

Case 2 Consider the generating function of the 2-variableLaguerre polynomials 119871

119899(119909 119910) [13]

1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905) =

infin

sum

119899=0

119871119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(44)

which can also be expressed as

exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119899(119909 119910)

119905119899

119899 (45)

where 1198620(119909) denotes the 0th-order Tricomi function [4]

Applying Theorem 1 to generating functions (44) and(45) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(46)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(47)

respectively

8 Journal of Mathematics

Taking 119910 = 1 in (46) and (47) and using the relation119871119899(119909 1) = 119871

119899(119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(48)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(49)

respectively where 119871119899(119909) are the ordinary Laguerre polyno-

mials [1]Again taking119910 = 0 in (46) and (47) and using the relation

119871119899(119909 0) = (minus119909)

119899

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(50)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(51)respectively

Next taking 119909 = 0 in (46) and (47) and using the relation119871119899(0 119910) = 119910

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590exp(minus 119887(119888 minus 119886)

2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) (119910119905)119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(52)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(119910119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(53)

respectively Replacing 119910 by minus119909 in (53) it reduces to (50)

Case 3 Consider the generating function of the Hermite-Appell polynomials

119867119860119899(1199091 1199092 1199093) [14 page 759 (23)]

119860 (119905) exp (1199091119905 + 119909

21199052

+ 11990931199053

) =

infin

sum

119899=0

119867119860119899(1199091 1199092 1199093)119905119899

119899

(54)

Applying Theorem 1 to generating function (54) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+

3

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119860119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(55)

Journal of Mathematics 9

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (55) and

using the relation119867119860119899(119909 0 0) = 119860

119899(119909) [14 page 760 (26)]

we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(56)

where 119860119899(119909) are the Appell polynomials [4 15]

Case 4 Consider the generating function of the Laguerre-Appell polynomials

119871119860119899(119909 119910) [16]

119860 (119905) [1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905)]

=

infin

sum

119899=0

119871119860119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(57)

which can also be expressed as follows

119860 (119905) exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119860119899(119909 119910)

119905119899

119899 (58)

Applying Theorem 1 to generating functions (57) and(58) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(59)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(60)

respectivelyTaking 119910 = 0 and replacing 119909 by minus119909 in (59) and (60) and

using the relation119871119860119899(119909 0) = 119860

119899(minus119909)119899 [16 page 9 (28)]

we get (56) and

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(61)

respectively

Case 5 Consider the generating function of the Hermite-Sheffer polynomials

119867119904119899(1199091 1199092 1199093) [17]

119860 (119905) exp (1199091119867(119905) + 119909

21198672

(119905) + 11990931198673

(119905))

=

infin

sum

119899=0

119867119904119899(1199091 1199092 1199093)119905119899

119899

(62)

Applying Theorem 1 to generating function (62) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

3

sum

119895=1

119909119895119867119895

(119905(119906 minus 119886)120588

(119888 minus 119906)120590

))119889119906

10 Journal of Mathematics

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119904119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(63)

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (63) and

using the relation119867119904119899(119909 0 0) = 119904

119899(119909) [17 page 16 (23)] we

get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119867 (119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119904119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(64)

where 119904119899(119909) are the Sheffer polynomials [15 18 19]

4 Concluding Remarks

In this paper we have derived the evaluations of certain Eulertype integralsWehave also established a theoremand appliedit to obtain evaluations of some integrals in terms of EBF119861(120572 120573 119887) We remark that these results can be extended tomultivariable case

To give an example we consider the following integralrepresentation [5 page 965 (20)] of Lauricellarsquos multiplehypergeometric series 119865(119899)

119863[4]

int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

119889119905

= 119861 (120572 120573) 119865(119899)

119863[120572 120572

1 1205722 120572

119899 120572 + 120573 119909

1 1199092 119909

119899]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(65)

Now taking 120601(119905) = prod119899

119894=1(1 minus 119909

119894119905)minus120572119894 120595(119905) = 1119905(1 minus 119905)

119886 = 0 and 119890 = 120574 = 1 in (15) using integral (65) in the rhs

after expanding119903119865119905[

(119891119903)

119889120595(119905)

(119892119905)

] in series and simplifying weget

119868119889120572120573101

[

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

1

119905 (1 minus 119905)]

=

infin

sum

11989811198982119898119899=0

(120572)1198981+1198982+sdotsdotsdot+119898

119899

(1205721)1198981

(1205722)1198982

sdot sdot sdot (120572119899)119898119899

(120572 + 120573)1198981+1198982+sdotsdotsdot+119898

119899

1199091198981

1

1198981

1199091198982

2

1198982sdot sdot sdot

119909119898119899

119899

119898119899

times119903+2119865119905+2

[[[[[[[

[

(119891119903) 1

2(1 minus 120572 minus 120573 minus

119899

sum

119894=1

119898119894) 1 minus

1

2(120572 + 120573 +

119899

sum

119894=1

119898119894)

4119889

(119892119905) 1 minus (120572 +

119899

sum

119894=1

119898119894) 1 minus 120573

]]]]]]]

]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(66)

We note that for 1205723= 120572

4= sdot sdot sdot = 0 (66) reduces to (17)

Further we extendTheorem 1 as follows

Theorem 5 Let the conditions for 119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

11989911989811198982119898119904=0

(119888 minus 119886)120583+(120588+120590)119899+119898

1+1198982+sdotsdotsdot+119898

119904minus1

times 119888119899119892119899(1199091 1199092 119909

119903)

times 119861(120582 + 120588119899 +

119904

sum

119894=1

119898119894 120583 minus 120582 + 120590119899 119887)

119904

prod

119894=1

(120572119894)119898119894

119910119898119894

119894

119898119894

119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(67)

Journal of Mathematics 11

The proof of this theorem is similar to that of Theorem 1The following are the consequences of Theorem 5

Corollary 6 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and119904 = 2 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

2

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899)

times 1198651[120582 + 120588119899 120572

1 1205722 120583 + (120588 + 120590) 119899

(119888 minus 119886) 1199101 (119888 minus 119886) 119910

2] 119905119899

(Re (120583) gt Re (120582) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(68)

Corollary 7 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and120588 = 120590 = 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905 (119906 minus 119886) (119888 minus 119906)) 119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904] 119905119899

(Re (120583) gt Re (120582) gt 0)

(69)

Example 8 Applying Corollary 7 to generating function (37)of the multivariable Hermite polynomials 119867(12119903)

119899(1199091 1199092

119909119903) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times exp(119903

sum

119895=1

119909119895(119905 (119906 minus 119886) (119888 minus 119906))

119895

)119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904]119905119899

119899

(Re (120583) gt Re (120582) gt 0)

(70)Remark 9 Taking 119886 = 0 and 119888 = 1 in (67) we get an extensionof integral (34)

Remark 10 Taking 119886 = minus1 and 119888 = 1 in (67) we get anextension of integral (35)

Clearly it appears that by applying Theorem 5 andCorollaries 6 and 7 to the generating functions defined by(37) (44) (45) (54) (57) (58) and (62) and also to someother generating functions a number of interesting integralscan be obtained

Further it is remarked that the operational methodsprovide useful tools to estimate specific theorems in variousfields of analysisThis approach requires the validation of anyof its consequences by a more rigorous procedure This isindeed the case of the Ramanujan master theorem [20 21]An operational method already employed to formulate ageneralization of the Ramanujan master theorem is appliedfor the evaluation of certain integrals by Babusci et al [22]

The technique adopted in [22] provides a very flexible andpowerful tool yielding new results encompassing differentaspects of the theory of special functions The possibility ofusing the method outlined in [22] for the integrals evaluatedin this article is a further research problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to thereviewer(s) for helpful suggestions and comments towardsthe improvement of this paper

References

[1] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985

[2] M A Chaudhry and S M Zubair ldquoGeneralized incompletegamma functions with applicationsrdquo Journal of Computationaland Applied Mathematics vol 55 no 1 pp 99ndash124 1994

[3] M A Chaudhry A Qadir M Rafique and S M ZubairldquoExtension of Eulerrsquos beta functionrdquo Journal of Computationaland Applied Mathematics vol 78 no 1 pp 19ndash32 1997

12 Journal of Mathematics

[4] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions John Wiley amp Sons New York NY USA 1984

[5] M E H Ismail and J Pitman ldquoAlgebraic evaluations of someEuler integrals duplication formulae for Appellrsquos hypergeomet-ric function 119865

1 and Brownian variationsrdquo Canadian Journal of

Mathematics vol 52 no 5 pp 961ndash981 2000[6] J Pitman ldquoBrownian motion bridge excursion and meander

characterized by sampling at independent uniform timesrdquoElectronic Journal of Probability vol 11 no 4 pp 1ndash33 1999

[7] Y A Brychkov Handbook of Special Functions DerivativesIntegrals Series and Other Formulas CRC Press New York NYUSA 2008

[8] G Dattoli S Lorenzutta G Maino A Torre and C CesaranoldquoGeneralized Hermite polynomials and super-Gaussian formsrdquoJournal of Mathematical Analysis and Applications vol 203 no3 pp 597ndash609 1996

[9] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[10] P Appell and J K de Feriet Fonctions Hypergeometriques etHyperspheriques Polynomes drsquoHermite Gauthier-Villars ParisFarnce 1926

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962

[12] M Lahiri ldquoOn a generalisation of Hermite polynomialsrdquoProceedings of the American Mathematical Society vol 27 pp117ndash121 1971

[13] G Dattoli and A Torre ldquoOperatorial methods and two variableLaguerre polynomialsrdquo Atti della Accademia delle Scienze diTorino Classe di Scienze Fisiche Matematiche e Naturali vol132 pp 1ndash7 1998

[14] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009

[15] S RomanTheUmbral Calculus Academic Press NewYorkNYUSA 1984

[16] S Khan M W Al-Saad and R Khan ldquoLaguerre-based Appellpolynomials properties and applicationsrdquo Mathematical andComputer Modelling vol 52 no 1-2 pp 247ndash259 2010

[17] S Khan M W Al-Saad and G Yasmin ldquoSome properties ofHermite-based Sheffer polynomialsrdquo Applied Mathematics andComputation vol 217 no 5 pp 2169ndash2183 2010

[18] E D Rainville Special Functions Macmillan New York NYUSA 1960

[19] E D Rainville SpecialFunctions Chelsea Publishing NewYorkNY USA 1971

[20] B C Berndt Ramanujanrsquos NotebooksmdashPart I Springer NewYork NY USA 1985

[21] G H Hardy Ramanujan Twelve Lectures on Subjects Suggestedby His Life and Work Cambridge University Press CambridgeUK 1940

[22] D Babusci G Dattoli G H E Duchamp K Gorska and K APenson ldquoDefinite integrals and operational methodsrdquo AppliedMathematics and Computation vol 219 no 6 pp 3017ndash30212012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Mathematics

=

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903) 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(34)

Proof Taking 119886 = 0 and 119888 = 1 in (32) we get (34)

Corollary 3 With definition (31) and notations as inTheorem 1 one has

int

1

minus1

(1 + 119906)120582minus1

(1 minus 119906)120583minus120582minus1

times 119866 (1199091 1199092 119909

119903 119905(1 + 119906)

120588

(1 minus 119906)120590

)

times exp(minus 4119887

(1 minus 1199062)) 119889119906

=

infin

sum

119899=0

(2)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0)

(35)

Proof Taking 119886 = minus1 and 119888 = 1 in (32) we get (35)

Corollary 4 With definition (31) and notations as inTheorem 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)minus120582minus1

119866(1199091 1199092 119909

119903 119905(

119888 minus 119906

119906 minus 119886)

120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=2

(119888 minus 119886) 1198902119887

infin

sum

119899=0

119888119899119892119899(1199091 1199092 119909

119903)119870

120582minus120590119899(2119887) 119905

119899

(Re (120582) gt 0Re (119887) gt 0)

(36)

Proof Taking 120583 = 0 and 120588 = minus120590 in (32) and making use ofrelation (10) we get (36)

Now we apply Theorem 1 to derive the evaluations ofcertain Euler type integrals in terms of EBF 119861(120572 120573 119887) Weconsider the following cases

Case 1 Consider the generating function of the multivariableHermite polynomials 119867(12119903)

119899(1199091 1199092 119909

119903) [8 page 602

(2021)]

exp (1199091119905 + 119909

21199052

+ sdot sdot sdot + 119909119903119905119903

)

=

infin

sum

119899=0

119867(12119903)

119899(1199091 1199092 119909

119903)119905119899

119899

(37)

Applying Theorem 1 to generating function (37) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

119903

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(38)

Taking 119903 = 3 in (38) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

3

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(39)

where119867119899(1199091 1199092 1199093) are the 3-variable Hermite polynomials

(3VHP) [9]Again taking 119903 = 2 in (38) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

2

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(1199091 1199092)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(40)

Journal of Mathematics 7

where 119867119899(1199091 1199092) are the 2-variable Hermite-Kampe de-

Feriet polynomials (2VHKdFP) [10]Next replacing 119909

1by 119909 119909

119903by 119910 and taking 119909

2=

1199093

= sdot sdot sdot = 119909119903minus1

= 0 in (38) and using the relation119867(12119903)

119899(119909 0 0 0 119910) = 119892

119903

119899(119909 119910) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

+119910(119905(119906 minus 119886)120588

(119888 minus 119906)120590

)119903

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119892119903

119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(41)

where 119892119903119899(119909 119910) are the Gould-Hopper generalized Hermite

polynomials (GHGHP) [11]Further replacing 119909

1by ]119909 and taking 119909

2= 119909

3= sdot sdot sdot =

119909119903minus1

= 0 119909119903= minus1 in (38) and using the relation119867(12119903)

119899(]119909

0 0 0 minus1) = 119867119899119903](119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+]119909119905(119906 minus 119886)120588(119888 minus 119906)120590minus (119905(119906 minus 119886)120588(119888 minus 119906)120590)119903)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899119903] (119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(42)

where119867119899119903](119909) are the generalizedHermite polynomials [12]

Furthermore taking 119903 = ] = 2 in (42) and using therelation119867

11989922(119909) = 119867

119899(119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+2119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

minus (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)2

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(43)

where119867119899(119909) are the ordinary Hermite polynomials [1]

Case 2 Consider the generating function of the 2-variableLaguerre polynomials 119871

119899(119909 119910) [13]

1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905) =

infin

sum

119899=0

119871119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(44)

which can also be expressed as

exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119899(119909 119910)

119905119899

119899 (45)

where 1198620(119909) denotes the 0th-order Tricomi function [4]

Applying Theorem 1 to generating functions (44) and(45) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(46)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(47)

respectively

8 Journal of Mathematics

Taking 119910 = 1 in (46) and (47) and using the relation119871119899(119909 1) = 119871

119899(119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(48)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(49)

respectively where 119871119899(119909) are the ordinary Laguerre polyno-

mials [1]Again taking119910 = 0 in (46) and (47) and using the relation

119871119899(119909 0) = (minus119909)

119899

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(50)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(51)respectively

Next taking 119909 = 0 in (46) and (47) and using the relation119871119899(0 119910) = 119910

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590exp(minus 119887(119888 minus 119886)

2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) (119910119905)119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(52)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(119910119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(53)

respectively Replacing 119910 by minus119909 in (53) it reduces to (50)

Case 3 Consider the generating function of the Hermite-Appell polynomials

119867119860119899(1199091 1199092 1199093) [14 page 759 (23)]

119860 (119905) exp (1199091119905 + 119909

21199052

+ 11990931199053

) =

infin

sum

119899=0

119867119860119899(1199091 1199092 1199093)119905119899

119899

(54)

Applying Theorem 1 to generating function (54) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+

3

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119860119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(55)

Journal of Mathematics 9

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (55) and

using the relation119867119860119899(119909 0 0) = 119860

119899(119909) [14 page 760 (26)]

we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(56)

where 119860119899(119909) are the Appell polynomials [4 15]

Case 4 Consider the generating function of the Laguerre-Appell polynomials

119871119860119899(119909 119910) [16]

119860 (119905) [1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905)]

=

infin

sum

119899=0

119871119860119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(57)

which can also be expressed as follows

119860 (119905) exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119860119899(119909 119910)

119905119899

119899 (58)

Applying Theorem 1 to generating functions (57) and(58) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(59)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(60)

respectivelyTaking 119910 = 0 and replacing 119909 by minus119909 in (59) and (60) and

using the relation119871119860119899(119909 0) = 119860

119899(minus119909)119899 [16 page 9 (28)]

we get (56) and

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(61)

respectively

Case 5 Consider the generating function of the Hermite-Sheffer polynomials

119867119904119899(1199091 1199092 1199093) [17]

119860 (119905) exp (1199091119867(119905) + 119909

21198672

(119905) + 11990931198673

(119905))

=

infin

sum

119899=0

119867119904119899(1199091 1199092 1199093)119905119899

119899

(62)

Applying Theorem 1 to generating function (62) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

3

sum

119895=1

119909119895119867119895

(119905(119906 minus 119886)120588

(119888 minus 119906)120590

))119889119906

10 Journal of Mathematics

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119904119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(63)

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (63) and

using the relation119867119904119899(119909 0 0) = 119904

119899(119909) [17 page 16 (23)] we

get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119867 (119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119904119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(64)

where 119904119899(119909) are the Sheffer polynomials [15 18 19]

4 Concluding Remarks

In this paper we have derived the evaluations of certain Eulertype integralsWehave also established a theoremand appliedit to obtain evaluations of some integrals in terms of EBF119861(120572 120573 119887) We remark that these results can be extended tomultivariable case

To give an example we consider the following integralrepresentation [5 page 965 (20)] of Lauricellarsquos multiplehypergeometric series 119865(119899)

119863[4]

int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

119889119905

= 119861 (120572 120573) 119865(119899)

119863[120572 120572

1 1205722 120572

119899 120572 + 120573 119909

1 1199092 119909

119899]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(65)

Now taking 120601(119905) = prod119899

119894=1(1 minus 119909

119894119905)minus120572119894 120595(119905) = 1119905(1 minus 119905)

119886 = 0 and 119890 = 120574 = 1 in (15) using integral (65) in the rhs

after expanding119903119865119905[

(119891119903)

119889120595(119905)

(119892119905)

] in series and simplifying weget

119868119889120572120573101

[

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

1

119905 (1 minus 119905)]

=

infin

sum

11989811198982119898119899=0

(120572)1198981+1198982+sdotsdotsdot+119898

119899

(1205721)1198981

(1205722)1198982

sdot sdot sdot (120572119899)119898119899

(120572 + 120573)1198981+1198982+sdotsdotsdot+119898

119899

1199091198981

1

1198981

1199091198982

2

1198982sdot sdot sdot

119909119898119899

119899

119898119899

times119903+2119865119905+2

[[[[[[[

[

(119891119903) 1

2(1 minus 120572 minus 120573 minus

119899

sum

119894=1

119898119894) 1 minus

1

2(120572 + 120573 +

119899

sum

119894=1

119898119894)

4119889

(119892119905) 1 minus (120572 +

119899

sum

119894=1

119898119894) 1 minus 120573

]]]]]]]

]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(66)

We note that for 1205723= 120572

4= sdot sdot sdot = 0 (66) reduces to (17)

Further we extendTheorem 1 as follows

Theorem 5 Let the conditions for 119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

11989911989811198982119898119904=0

(119888 minus 119886)120583+(120588+120590)119899+119898

1+1198982+sdotsdotsdot+119898

119904minus1

times 119888119899119892119899(1199091 1199092 119909

119903)

times 119861(120582 + 120588119899 +

119904

sum

119894=1

119898119894 120583 minus 120582 + 120590119899 119887)

119904

prod

119894=1

(120572119894)119898119894

119910119898119894

119894

119898119894

119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(67)

Journal of Mathematics 11

The proof of this theorem is similar to that of Theorem 1The following are the consequences of Theorem 5

Corollary 6 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and119904 = 2 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

2

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899)

times 1198651[120582 + 120588119899 120572

1 1205722 120583 + (120588 + 120590) 119899

(119888 minus 119886) 1199101 (119888 minus 119886) 119910

2] 119905119899

(Re (120583) gt Re (120582) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(68)

Corollary 7 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and120588 = 120590 = 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905 (119906 minus 119886) (119888 minus 119906)) 119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904] 119905119899

(Re (120583) gt Re (120582) gt 0)

(69)

Example 8 Applying Corollary 7 to generating function (37)of the multivariable Hermite polynomials 119867(12119903)

119899(1199091 1199092

119909119903) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times exp(119903

sum

119895=1

119909119895(119905 (119906 minus 119886) (119888 minus 119906))

119895

)119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904]119905119899

119899

(Re (120583) gt Re (120582) gt 0)

(70)Remark 9 Taking 119886 = 0 and 119888 = 1 in (67) we get an extensionof integral (34)

Remark 10 Taking 119886 = minus1 and 119888 = 1 in (67) we get anextension of integral (35)

Clearly it appears that by applying Theorem 5 andCorollaries 6 and 7 to the generating functions defined by(37) (44) (45) (54) (57) (58) and (62) and also to someother generating functions a number of interesting integralscan be obtained

Further it is remarked that the operational methodsprovide useful tools to estimate specific theorems in variousfields of analysisThis approach requires the validation of anyof its consequences by a more rigorous procedure This isindeed the case of the Ramanujan master theorem [20 21]An operational method already employed to formulate ageneralization of the Ramanujan master theorem is appliedfor the evaluation of certain integrals by Babusci et al [22]

The technique adopted in [22] provides a very flexible andpowerful tool yielding new results encompassing differentaspects of the theory of special functions The possibility ofusing the method outlined in [22] for the integrals evaluatedin this article is a further research problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to thereviewer(s) for helpful suggestions and comments towardsthe improvement of this paper

References

[1] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985

[2] M A Chaudhry and S M Zubair ldquoGeneralized incompletegamma functions with applicationsrdquo Journal of Computationaland Applied Mathematics vol 55 no 1 pp 99ndash124 1994

[3] M A Chaudhry A Qadir M Rafique and S M ZubairldquoExtension of Eulerrsquos beta functionrdquo Journal of Computationaland Applied Mathematics vol 78 no 1 pp 19ndash32 1997

12 Journal of Mathematics

[4] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions John Wiley amp Sons New York NY USA 1984

[5] M E H Ismail and J Pitman ldquoAlgebraic evaluations of someEuler integrals duplication formulae for Appellrsquos hypergeomet-ric function 119865

1 and Brownian variationsrdquo Canadian Journal of

Mathematics vol 52 no 5 pp 961ndash981 2000[6] J Pitman ldquoBrownian motion bridge excursion and meander

characterized by sampling at independent uniform timesrdquoElectronic Journal of Probability vol 11 no 4 pp 1ndash33 1999

[7] Y A Brychkov Handbook of Special Functions DerivativesIntegrals Series and Other Formulas CRC Press New York NYUSA 2008

[8] G Dattoli S Lorenzutta G Maino A Torre and C CesaranoldquoGeneralized Hermite polynomials and super-Gaussian formsrdquoJournal of Mathematical Analysis and Applications vol 203 no3 pp 597ndash609 1996

[9] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[10] P Appell and J K de Feriet Fonctions Hypergeometriques etHyperspheriques Polynomes drsquoHermite Gauthier-Villars ParisFarnce 1926

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962

[12] M Lahiri ldquoOn a generalisation of Hermite polynomialsrdquoProceedings of the American Mathematical Society vol 27 pp117ndash121 1971

[13] G Dattoli and A Torre ldquoOperatorial methods and two variableLaguerre polynomialsrdquo Atti della Accademia delle Scienze diTorino Classe di Scienze Fisiche Matematiche e Naturali vol132 pp 1ndash7 1998

[14] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009

[15] S RomanTheUmbral Calculus Academic Press NewYorkNYUSA 1984

[16] S Khan M W Al-Saad and R Khan ldquoLaguerre-based Appellpolynomials properties and applicationsrdquo Mathematical andComputer Modelling vol 52 no 1-2 pp 247ndash259 2010

[17] S Khan M W Al-Saad and G Yasmin ldquoSome properties ofHermite-based Sheffer polynomialsrdquo Applied Mathematics andComputation vol 217 no 5 pp 2169ndash2183 2010

[18] E D Rainville Special Functions Macmillan New York NYUSA 1960

[19] E D Rainville SpecialFunctions Chelsea Publishing NewYorkNY USA 1971

[20] B C Berndt Ramanujanrsquos NotebooksmdashPart I Springer NewYork NY USA 1985

[21] G H Hardy Ramanujan Twelve Lectures on Subjects Suggestedby His Life and Work Cambridge University Press CambridgeUK 1940

[22] D Babusci G Dattoli G H E Duchamp K Gorska and K APenson ldquoDefinite integrals and operational methodsrdquo AppliedMathematics and Computation vol 219 no 6 pp 3017ndash30212012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 7

where 119867119899(1199091 1199092) are the 2-variable Hermite-Kampe de-

Feriet polynomials (2VHKdFP) [10]Next replacing 119909

1by 119909 119909

119903by 119910 and taking 119909

2=

1199093

= sdot sdot sdot = 119909119903minus1

= 0 in (38) and using the relation119867(12119903)

119899(119909 0 0 0 119910) = 119892

119903

119899(119909 119910) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

+119910(119905(119906 minus 119886)120588

(119888 minus 119906)120590

)119903

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119892119903

119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(41)

where 119892119903119899(119909 119910) are the Gould-Hopper generalized Hermite

polynomials (GHGHP) [11]Further replacing 119909

1by ]119909 and taking 119909

2= 119909

3= sdot sdot sdot =

119909119903minus1

= 0 119909119903= minus1 in (38) and using the relation119867(12119903)

119899(]119909

0 0 0 minus1) = 119867119899119903](119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+]119909119905(119906 minus 119886)120588(119888 minus 119906)120590minus (119905(119906 minus 119886)120588(119888 minus 119906)120590)119903)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899119903] (119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(42)

where119867119899119903](119909) are the generalizedHermite polynomials [12]

Furthermore taking 119903 = ] = 2 in (42) and using therelation119867

11989922(119909) = 119867

119899(119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+2119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

minus (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)2

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(43)

where119867119899(119909) are the ordinary Hermite polynomials [1]

Case 2 Consider the generating function of the 2-variableLaguerre polynomials 119871

119899(119909 119910) [13]

1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905) =

infin

sum

119899=0

119871119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(44)

which can also be expressed as

exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119899(119909 119910)

119905119899

119899 (45)

where 1198620(119909) denotes the 0th-order Tricomi function [4]

Applying Theorem 1 to generating functions (44) and(45) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(46)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(47)

respectively

8 Journal of Mathematics

Taking 119910 = 1 in (46) and (47) and using the relation119871119899(119909 1) = 119871

119899(119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(48)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(49)

respectively where 119871119899(119909) are the ordinary Laguerre polyno-

mials [1]Again taking119910 = 0 in (46) and (47) and using the relation

119871119899(119909 0) = (minus119909)

119899

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(50)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(51)respectively

Next taking 119909 = 0 in (46) and (47) and using the relation119871119899(0 119910) = 119910

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590exp(minus 119887(119888 minus 119886)

2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) (119910119905)119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(52)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(119910119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(53)

respectively Replacing 119910 by minus119909 in (53) it reduces to (50)

Case 3 Consider the generating function of the Hermite-Appell polynomials

119867119860119899(1199091 1199092 1199093) [14 page 759 (23)]

119860 (119905) exp (1199091119905 + 119909

21199052

+ 11990931199053

) =

infin

sum

119899=0

119867119860119899(1199091 1199092 1199093)119905119899

119899

(54)

Applying Theorem 1 to generating function (54) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+

3

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119860119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(55)

Journal of Mathematics 9

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (55) and

using the relation119867119860119899(119909 0 0) = 119860

119899(119909) [14 page 760 (26)]

we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(56)

where 119860119899(119909) are the Appell polynomials [4 15]

Case 4 Consider the generating function of the Laguerre-Appell polynomials

119871119860119899(119909 119910) [16]

119860 (119905) [1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905)]

=

infin

sum

119899=0

119871119860119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(57)

which can also be expressed as follows

119860 (119905) exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119860119899(119909 119910)

119905119899

119899 (58)

Applying Theorem 1 to generating functions (57) and(58) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(59)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(60)

respectivelyTaking 119910 = 0 and replacing 119909 by minus119909 in (59) and (60) and

using the relation119871119860119899(119909 0) = 119860

119899(minus119909)119899 [16 page 9 (28)]

we get (56) and

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(61)

respectively

Case 5 Consider the generating function of the Hermite-Sheffer polynomials

119867119904119899(1199091 1199092 1199093) [17]

119860 (119905) exp (1199091119867(119905) + 119909

21198672

(119905) + 11990931198673

(119905))

=

infin

sum

119899=0

119867119904119899(1199091 1199092 1199093)119905119899

119899

(62)

Applying Theorem 1 to generating function (62) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

3

sum

119895=1

119909119895119867119895

(119905(119906 minus 119886)120588

(119888 minus 119906)120590

))119889119906

10 Journal of Mathematics

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119904119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(63)

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (63) and

using the relation119867119904119899(119909 0 0) = 119904

119899(119909) [17 page 16 (23)] we

get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119867 (119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119904119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(64)

where 119904119899(119909) are the Sheffer polynomials [15 18 19]

4 Concluding Remarks

In this paper we have derived the evaluations of certain Eulertype integralsWehave also established a theoremand appliedit to obtain evaluations of some integrals in terms of EBF119861(120572 120573 119887) We remark that these results can be extended tomultivariable case

To give an example we consider the following integralrepresentation [5 page 965 (20)] of Lauricellarsquos multiplehypergeometric series 119865(119899)

119863[4]

int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

119889119905

= 119861 (120572 120573) 119865(119899)

119863[120572 120572

1 1205722 120572

119899 120572 + 120573 119909

1 1199092 119909

119899]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(65)

Now taking 120601(119905) = prod119899

119894=1(1 minus 119909

119894119905)minus120572119894 120595(119905) = 1119905(1 minus 119905)

119886 = 0 and 119890 = 120574 = 1 in (15) using integral (65) in the rhs

after expanding119903119865119905[

(119891119903)

119889120595(119905)

(119892119905)

] in series and simplifying weget

119868119889120572120573101

[

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

1

119905 (1 minus 119905)]

=

infin

sum

11989811198982119898119899=0

(120572)1198981+1198982+sdotsdotsdot+119898

119899

(1205721)1198981

(1205722)1198982

sdot sdot sdot (120572119899)119898119899

(120572 + 120573)1198981+1198982+sdotsdotsdot+119898

119899

1199091198981

1

1198981

1199091198982

2

1198982sdot sdot sdot

119909119898119899

119899

119898119899

times119903+2119865119905+2

[[[[[[[

[

(119891119903) 1

2(1 minus 120572 minus 120573 minus

119899

sum

119894=1

119898119894) 1 minus

1

2(120572 + 120573 +

119899

sum

119894=1

119898119894)

4119889

(119892119905) 1 minus (120572 +

119899

sum

119894=1

119898119894) 1 minus 120573

]]]]]]]

]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(66)

We note that for 1205723= 120572

4= sdot sdot sdot = 0 (66) reduces to (17)

Further we extendTheorem 1 as follows

Theorem 5 Let the conditions for 119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

11989911989811198982119898119904=0

(119888 minus 119886)120583+(120588+120590)119899+119898

1+1198982+sdotsdotsdot+119898

119904minus1

times 119888119899119892119899(1199091 1199092 119909

119903)

times 119861(120582 + 120588119899 +

119904

sum

119894=1

119898119894 120583 minus 120582 + 120590119899 119887)

119904

prod

119894=1

(120572119894)119898119894

119910119898119894

119894

119898119894

119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(67)

Journal of Mathematics 11

The proof of this theorem is similar to that of Theorem 1The following are the consequences of Theorem 5

Corollary 6 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and119904 = 2 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

2

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899)

times 1198651[120582 + 120588119899 120572

1 1205722 120583 + (120588 + 120590) 119899

(119888 minus 119886) 1199101 (119888 minus 119886) 119910

2] 119905119899

(Re (120583) gt Re (120582) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(68)

Corollary 7 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and120588 = 120590 = 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905 (119906 minus 119886) (119888 minus 119906)) 119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904] 119905119899

(Re (120583) gt Re (120582) gt 0)

(69)

Example 8 Applying Corollary 7 to generating function (37)of the multivariable Hermite polynomials 119867(12119903)

119899(1199091 1199092

119909119903) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times exp(119903

sum

119895=1

119909119895(119905 (119906 minus 119886) (119888 minus 119906))

119895

)119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904]119905119899

119899

(Re (120583) gt Re (120582) gt 0)

(70)Remark 9 Taking 119886 = 0 and 119888 = 1 in (67) we get an extensionof integral (34)

Remark 10 Taking 119886 = minus1 and 119888 = 1 in (67) we get anextension of integral (35)

Clearly it appears that by applying Theorem 5 andCorollaries 6 and 7 to the generating functions defined by(37) (44) (45) (54) (57) (58) and (62) and also to someother generating functions a number of interesting integralscan be obtained

Further it is remarked that the operational methodsprovide useful tools to estimate specific theorems in variousfields of analysisThis approach requires the validation of anyof its consequences by a more rigorous procedure This isindeed the case of the Ramanujan master theorem [20 21]An operational method already employed to formulate ageneralization of the Ramanujan master theorem is appliedfor the evaluation of certain integrals by Babusci et al [22]

The technique adopted in [22] provides a very flexible andpowerful tool yielding new results encompassing differentaspects of the theory of special functions The possibility ofusing the method outlined in [22] for the integrals evaluatedin this article is a further research problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to thereviewer(s) for helpful suggestions and comments towardsthe improvement of this paper

References

[1] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985

[2] M A Chaudhry and S M Zubair ldquoGeneralized incompletegamma functions with applicationsrdquo Journal of Computationaland Applied Mathematics vol 55 no 1 pp 99ndash124 1994

[3] M A Chaudhry A Qadir M Rafique and S M ZubairldquoExtension of Eulerrsquos beta functionrdquo Journal of Computationaland Applied Mathematics vol 78 no 1 pp 19ndash32 1997

12 Journal of Mathematics

[4] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions John Wiley amp Sons New York NY USA 1984

[5] M E H Ismail and J Pitman ldquoAlgebraic evaluations of someEuler integrals duplication formulae for Appellrsquos hypergeomet-ric function 119865

1 and Brownian variationsrdquo Canadian Journal of

Mathematics vol 52 no 5 pp 961ndash981 2000[6] J Pitman ldquoBrownian motion bridge excursion and meander

characterized by sampling at independent uniform timesrdquoElectronic Journal of Probability vol 11 no 4 pp 1ndash33 1999

[7] Y A Brychkov Handbook of Special Functions DerivativesIntegrals Series and Other Formulas CRC Press New York NYUSA 2008

[8] G Dattoli S Lorenzutta G Maino A Torre and C CesaranoldquoGeneralized Hermite polynomials and super-Gaussian formsrdquoJournal of Mathematical Analysis and Applications vol 203 no3 pp 597ndash609 1996

[9] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[10] P Appell and J K de Feriet Fonctions Hypergeometriques etHyperspheriques Polynomes drsquoHermite Gauthier-Villars ParisFarnce 1926

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962

[12] M Lahiri ldquoOn a generalisation of Hermite polynomialsrdquoProceedings of the American Mathematical Society vol 27 pp117ndash121 1971

[13] G Dattoli and A Torre ldquoOperatorial methods and two variableLaguerre polynomialsrdquo Atti della Accademia delle Scienze diTorino Classe di Scienze Fisiche Matematiche e Naturali vol132 pp 1ndash7 1998

[14] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009

[15] S RomanTheUmbral Calculus Academic Press NewYorkNYUSA 1984

[16] S Khan M W Al-Saad and R Khan ldquoLaguerre-based Appellpolynomials properties and applicationsrdquo Mathematical andComputer Modelling vol 52 no 1-2 pp 247ndash259 2010

[17] S Khan M W Al-Saad and G Yasmin ldquoSome properties ofHermite-based Sheffer polynomialsrdquo Applied Mathematics andComputation vol 217 no 5 pp 2169ndash2183 2010

[18] E D Rainville Special Functions Macmillan New York NYUSA 1960

[19] E D Rainville SpecialFunctions Chelsea Publishing NewYorkNY USA 1971

[20] B C Berndt Ramanujanrsquos NotebooksmdashPart I Springer NewYork NY USA 1985

[21] G H Hardy Ramanujan Twelve Lectures on Subjects Suggestedby His Life and Work Cambridge University Press CambridgeUK 1940

[22] D Babusci G Dattoli G H E Duchamp K Gorska and K APenson ldquoDefinite integrals and operational methodsrdquo AppliedMathematics and Computation vol 219 no 6 pp 3017ndash30212012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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8 Journal of Mathematics

Taking 119910 = 1 in (46) and (47) and using the relation119871119899(119909 1) = 119871

119899(119909) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(48)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(49)

respectively where 119871119899(119909) are the ordinary Laguerre polyno-

mials [1]Again taking119910 = 0 in (46) and (47) and using the relation

119871119899(119909 0) = (minus119909)

119899

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(50)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(minus119909119905)

119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(51)respectively

Next taking 119909 = 0 in (46) and (47) and using the relation119871119899(0 119910) = 119910

119899 we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590exp(minus 119887(119888 minus 119886)

2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) (119910119905)119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(52)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)(119910119905)

119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(53)

respectively Replacing 119910 by minus119909 in (53) it reduces to (50)

Case 3 Consider the generating function of the Hermite-Appell polynomials

119867119860119899(1199091 1199092 1199093) [14 page 759 (23)]

119860 (119905) exp (1199091119905 + 119909

21199052

+ 11990931199053

) =

infin

sum

119899=0

119867119860119899(1199091 1199092 1199093)119905119899

119899

(54)

Applying Theorem 1 to generating function (54) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+

3

sum

119895=1

119909119895(119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119895

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119860119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(55)

Journal of Mathematics 9

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (55) and

using the relation119867119860119899(119909 0 0) = 119860

119899(119909) [14 page 760 (26)]

we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(56)

where 119860119899(119909) are the Appell polynomials [4 15]

Case 4 Consider the generating function of the Laguerre-Appell polynomials

119871119860119899(119909 119910) [16]

119860 (119905) [1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905)]

=

infin

sum

119899=0

119871119860119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(57)

which can also be expressed as follows

119860 (119905) exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119860119899(119909 119910)

119905119899

119899 (58)

Applying Theorem 1 to generating functions (57) and(58) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(59)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(60)

respectivelyTaking 119910 = 0 and replacing 119909 by minus119909 in (59) and (60) and

using the relation119871119860119899(119909 0) = 119860

119899(minus119909)119899 [16 page 9 (28)]

we get (56) and

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(61)

respectively

Case 5 Consider the generating function of the Hermite-Sheffer polynomials

119867119904119899(1199091 1199092 1199093) [17]

119860 (119905) exp (1199091119867(119905) + 119909

21198672

(119905) + 11990931198673

(119905))

=

infin

sum

119899=0

119867119904119899(1199091 1199092 1199093)119905119899

119899

(62)

Applying Theorem 1 to generating function (62) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

3

sum

119895=1

119909119895119867119895

(119905(119906 minus 119886)120588

(119888 minus 119906)120590

))119889119906

10 Journal of Mathematics

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119904119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(63)

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (63) and

using the relation119867119904119899(119909 0 0) = 119904

119899(119909) [17 page 16 (23)] we

get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119867 (119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119904119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(64)

where 119904119899(119909) are the Sheffer polynomials [15 18 19]

4 Concluding Remarks

In this paper we have derived the evaluations of certain Eulertype integralsWehave also established a theoremand appliedit to obtain evaluations of some integrals in terms of EBF119861(120572 120573 119887) We remark that these results can be extended tomultivariable case

To give an example we consider the following integralrepresentation [5 page 965 (20)] of Lauricellarsquos multiplehypergeometric series 119865(119899)

119863[4]

int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

119889119905

= 119861 (120572 120573) 119865(119899)

119863[120572 120572

1 1205722 120572

119899 120572 + 120573 119909

1 1199092 119909

119899]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(65)

Now taking 120601(119905) = prod119899

119894=1(1 minus 119909

119894119905)minus120572119894 120595(119905) = 1119905(1 minus 119905)

119886 = 0 and 119890 = 120574 = 1 in (15) using integral (65) in the rhs

after expanding119903119865119905[

(119891119903)

119889120595(119905)

(119892119905)

] in series and simplifying weget

119868119889120572120573101

[

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

1

119905 (1 minus 119905)]

=

infin

sum

11989811198982119898119899=0

(120572)1198981+1198982+sdotsdotsdot+119898

119899

(1205721)1198981

(1205722)1198982

sdot sdot sdot (120572119899)119898119899

(120572 + 120573)1198981+1198982+sdotsdotsdot+119898

119899

1199091198981

1

1198981

1199091198982

2

1198982sdot sdot sdot

119909119898119899

119899

119898119899

times119903+2119865119905+2

[[[[[[[

[

(119891119903) 1

2(1 minus 120572 minus 120573 minus

119899

sum

119894=1

119898119894) 1 minus

1

2(120572 + 120573 +

119899

sum

119894=1

119898119894)

4119889

(119892119905) 1 minus (120572 +

119899

sum

119894=1

119898119894) 1 minus 120573

]]]]]]]

]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(66)

We note that for 1205723= 120572

4= sdot sdot sdot = 0 (66) reduces to (17)

Further we extendTheorem 1 as follows

Theorem 5 Let the conditions for 119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

11989911989811198982119898119904=0

(119888 minus 119886)120583+(120588+120590)119899+119898

1+1198982+sdotsdotsdot+119898

119904minus1

times 119888119899119892119899(1199091 1199092 119909

119903)

times 119861(120582 + 120588119899 +

119904

sum

119894=1

119898119894 120583 minus 120582 + 120590119899 119887)

119904

prod

119894=1

(120572119894)119898119894

119910119898119894

119894

119898119894

119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(67)

Journal of Mathematics 11

The proof of this theorem is similar to that of Theorem 1The following are the consequences of Theorem 5

Corollary 6 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and119904 = 2 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

2

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899)

times 1198651[120582 + 120588119899 120572

1 1205722 120583 + (120588 + 120590) 119899

(119888 minus 119886) 1199101 (119888 minus 119886) 119910

2] 119905119899

(Re (120583) gt Re (120582) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(68)

Corollary 7 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and120588 = 120590 = 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905 (119906 minus 119886) (119888 minus 119906)) 119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904] 119905119899

(Re (120583) gt Re (120582) gt 0)

(69)

Example 8 Applying Corollary 7 to generating function (37)of the multivariable Hermite polynomials 119867(12119903)

119899(1199091 1199092

119909119903) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times exp(119903

sum

119895=1

119909119895(119905 (119906 minus 119886) (119888 minus 119906))

119895

)119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904]119905119899

119899

(Re (120583) gt Re (120582) gt 0)

(70)Remark 9 Taking 119886 = 0 and 119888 = 1 in (67) we get an extensionof integral (34)

Remark 10 Taking 119886 = minus1 and 119888 = 1 in (67) we get anextension of integral (35)

Clearly it appears that by applying Theorem 5 andCorollaries 6 and 7 to the generating functions defined by(37) (44) (45) (54) (57) (58) and (62) and also to someother generating functions a number of interesting integralscan be obtained

Further it is remarked that the operational methodsprovide useful tools to estimate specific theorems in variousfields of analysisThis approach requires the validation of anyof its consequences by a more rigorous procedure This isindeed the case of the Ramanujan master theorem [20 21]An operational method already employed to formulate ageneralization of the Ramanujan master theorem is appliedfor the evaluation of certain integrals by Babusci et al [22]

The technique adopted in [22] provides a very flexible andpowerful tool yielding new results encompassing differentaspects of the theory of special functions The possibility ofusing the method outlined in [22] for the integrals evaluatedin this article is a further research problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to thereviewer(s) for helpful suggestions and comments towardsthe improvement of this paper

References

[1] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985

[2] M A Chaudhry and S M Zubair ldquoGeneralized incompletegamma functions with applicationsrdquo Journal of Computationaland Applied Mathematics vol 55 no 1 pp 99ndash124 1994

[3] M A Chaudhry A Qadir M Rafique and S M ZubairldquoExtension of Eulerrsquos beta functionrdquo Journal of Computationaland Applied Mathematics vol 78 no 1 pp 19ndash32 1997

12 Journal of Mathematics

[4] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions John Wiley amp Sons New York NY USA 1984

[5] M E H Ismail and J Pitman ldquoAlgebraic evaluations of someEuler integrals duplication formulae for Appellrsquos hypergeomet-ric function 119865

1 and Brownian variationsrdquo Canadian Journal of

Mathematics vol 52 no 5 pp 961ndash981 2000[6] J Pitman ldquoBrownian motion bridge excursion and meander

characterized by sampling at independent uniform timesrdquoElectronic Journal of Probability vol 11 no 4 pp 1ndash33 1999

[7] Y A Brychkov Handbook of Special Functions DerivativesIntegrals Series and Other Formulas CRC Press New York NYUSA 2008

[8] G Dattoli S Lorenzutta G Maino A Torre and C CesaranoldquoGeneralized Hermite polynomials and super-Gaussian formsrdquoJournal of Mathematical Analysis and Applications vol 203 no3 pp 597ndash609 1996

[9] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[10] P Appell and J K de Feriet Fonctions Hypergeometriques etHyperspheriques Polynomes drsquoHermite Gauthier-Villars ParisFarnce 1926

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962

[12] M Lahiri ldquoOn a generalisation of Hermite polynomialsrdquoProceedings of the American Mathematical Society vol 27 pp117ndash121 1971

[13] G Dattoli and A Torre ldquoOperatorial methods and two variableLaguerre polynomialsrdquo Atti della Accademia delle Scienze diTorino Classe di Scienze Fisiche Matematiche e Naturali vol132 pp 1ndash7 1998

[14] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009

[15] S RomanTheUmbral Calculus Academic Press NewYorkNYUSA 1984

[16] S Khan M W Al-Saad and R Khan ldquoLaguerre-based Appellpolynomials properties and applicationsrdquo Mathematical andComputer Modelling vol 52 no 1-2 pp 247ndash259 2010

[17] S Khan M W Al-Saad and G Yasmin ldquoSome properties ofHermite-based Sheffer polynomialsrdquo Applied Mathematics andComputation vol 217 no 5 pp 2169ndash2183 2010

[18] E D Rainville Special Functions Macmillan New York NYUSA 1960

[19] E D Rainville SpecialFunctions Chelsea Publishing NewYorkNY USA 1971

[20] B C Berndt Ramanujanrsquos NotebooksmdashPart I Springer NewYork NY USA 1985

[21] G H Hardy Ramanujan Twelve Lectures on Subjects Suggestedby His Life and Work Cambridge University Press CambridgeUK 1940

[22] D Babusci G Dattoli G H E Duchamp K Gorska and K APenson ldquoDefinite integrals and operational methodsrdquo AppliedMathematics and Computation vol 219 no 6 pp 3017ndash30212012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 9

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (55) and

using the relation119867119860119899(119909 0 0) = 119860

119899(119909) [14 page 760 (26)]

we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(56)

where 119860119899(119909) are the Appell polynomials [4 15]

Case 4 Consider the generating function of the Laguerre-Appell polynomials

119871119860119899(119909 119910) [16]

119860 (119905) [1

(1 minus 119910119905)exp( minus119909119905

1 minus 119910119905)]

=

infin

sum

119899=0

119871119860119899(119909 119910) 119905

119899

(1003816100381610038161003816119910119905

1003816100381610038161003816 lt 1)

(57)

which can also be expressed as follows

119860 (119905) exp (119910119905) 1198620(119909119905) =

infin

sum

119899=0

119871119860119899(119909 119910)

119905119899

119899 (58)

Applying Theorem 1 to generating functions (57) and(58) we find the following integrals

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)minus

119909119905(119906 minus 119886)120588

(119888 minus 119906)120590

1 minus 119910119905(119906 minus 119886)120588

(119888 minus 119906)120590)119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887) 119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0

1003816100381610038161003816119910119905(119906 minus 119886)120588

(119888 minus 119906)1205901003816100381610038161003816 lt 1 120588 120590 ge 0 120588 + 120590 gt 0)

(59)

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119910119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times 1198620(119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119871119860119899(119909 119910)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(60)

respectivelyTaking 119910 = 0 and replacing 119909 by minus119909 in (59) and (60) and

using the relation119871119860119899(119909 0) = 119860

119899(minus119909)119899 [16 page 9 (28)]

we get (56) and

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906))119862

0(minus119909119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119860119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

(119899)2

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(61)

respectively

Case 5 Consider the generating function of the Hermite-Sheffer polynomials

119867119904119899(1199091 1199092 1199093) [17]

119860 (119905) exp (1199091119867(119905) + 119909

21198672

(119905) + 11990931198673

(119905))

=

infin

sum

119899=0

119867119904119899(1199091 1199092 1199093)119905119899

119899

(62)

Applying Theorem 1 to generating function (62) we findthe following integral

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)

+

3

sum

119895=1

119909119895119867119895

(119905(119906 minus 119886)120588

(119888 minus 119906)120590

))119889119906

10 Journal of Mathematics

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119904119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(63)

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (63) and

using the relation119867119904119899(119909 0 0) = 119904

119899(119909) [17 page 16 (23)] we

get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119867 (119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119904119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(64)

where 119904119899(119909) are the Sheffer polynomials [15 18 19]

4 Concluding Remarks

In this paper we have derived the evaluations of certain Eulertype integralsWehave also established a theoremand appliedit to obtain evaluations of some integrals in terms of EBF119861(120572 120573 119887) We remark that these results can be extended tomultivariable case

To give an example we consider the following integralrepresentation [5 page 965 (20)] of Lauricellarsquos multiplehypergeometric series 119865(119899)

119863[4]

int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

119889119905

= 119861 (120572 120573) 119865(119899)

119863[120572 120572

1 1205722 120572

119899 120572 + 120573 119909

1 1199092 119909

119899]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(65)

Now taking 120601(119905) = prod119899

119894=1(1 minus 119909

119894119905)minus120572119894 120595(119905) = 1119905(1 minus 119905)

119886 = 0 and 119890 = 120574 = 1 in (15) using integral (65) in the rhs

after expanding119903119865119905[

(119891119903)

119889120595(119905)

(119892119905)

] in series and simplifying weget

119868119889120572120573101

[

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

1

119905 (1 minus 119905)]

=

infin

sum

11989811198982119898119899=0

(120572)1198981+1198982+sdotsdotsdot+119898

119899

(1205721)1198981

(1205722)1198982

sdot sdot sdot (120572119899)119898119899

(120572 + 120573)1198981+1198982+sdotsdotsdot+119898

119899

1199091198981

1

1198981

1199091198982

2

1198982sdot sdot sdot

119909119898119899

119899

119898119899

times119903+2119865119905+2

[[[[[[[

[

(119891119903) 1

2(1 minus 120572 minus 120573 minus

119899

sum

119894=1

119898119894) 1 minus

1

2(120572 + 120573 +

119899

sum

119894=1

119898119894)

4119889

(119892119905) 1 minus (120572 +

119899

sum

119894=1

119898119894) 1 minus 120573

]]]]]]]

]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(66)

We note that for 1205723= 120572

4= sdot sdot sdot = 0 (66) reduces to (17)

Further we extendTheorem 1 as follows

Theorem 5 Let the conditions for 119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

11989911989811198982119898119904=0

(119888 minus 119886)120583+(120588+120590)119899+119898

1+1198982+sdotsdotsdot+119898

119904minus1

times 119888119899119892119899(1199091 1199092 119909

119903)

times 119861(120582 + 120588119899 +

119904

sum

119894=1

119898119894 120583 minus 120582 + 120590119899 119887)

119904

prod

119894=1

(120572119894)119898119894

119910119898119894

119894

119898119894

119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(67)

Journal of Mathematics 11

The proof of this theorem is similar to that of Theorem 1The following are the consequences of Theorem 5

Corollary 6 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and119904 = 2 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

2

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899)

times 1198651[120582 + 120588119899 120572

1 1205722 120583 + (120588 + 120590) 119899

(119888 minus 119886) 1199101 (119888 minus 119886) 119910

2] 119905119899

(Re (120583) gt Re (120582) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(68)

Corollary 7 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and120588 = 120590 = 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905 (119906 minus 119886) (119888 minus 119906)) 119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904] 119905119899

(Re (120583) gt Re (120582) gt 0)

(69)

Example 8 Applying Corollary 7 to generating function (37)of the multivariable Hermite polynomials 119867(12119903)

119899(1199091 1199092

119909119903) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times exp(119903

sum

119895=1

119909119895(119905 (119906 minus 119886) (119888 minus 119906))

119895

)119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904]119905119899

119899

(Re (120583) gt Re (120582) gt 0)

(70)Remark 9 Taking 119886 = 0 and 119888 = 1 in (67) we get an extensionof integral (34)

Remark 10 Taking 119886 = minus1 and 119888 = 1 in (67) we get anextension of integral (35)

Clearly it appears that by applying Theorem 5 andCorollaries 6 and 7 to the generating functions defined by(37) (44) (45) (54) (57) (58) and (62) and also to someother generating functions a number of interesting integralscan be obtained

Further it is remarked that the operational methodsprovide useful tools to estimate specific theorems in variousfields of analysisThis approach requires the validation of anyof its consequences by a more rigorous procedure This isindeed the case of the Ramanujan master theorem [20 21]An operational method already employed to formulate ageneralization of the Ramanujan master theorem is appliedfor the evaluation of certain integrals by Babusci et al [22]

The technique adopted in [22] provides a very flexible andpowerful tool yielding new results encompassing differentaspects of the theory of special functions The possibility ofusing the method outlined in [22] for the integrals evaluatedin this article is a further research problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to thereviewer(s) for helpful suggestions and comments towardsthe improvement of this paper

References

[1] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985

[2] M A Chaudhry and S M Zubair ldquoGeneralized incompletegamma functions with applicationsrdquo Journal of Computationaland Applied Mathematics vol 55 no 1 pp 99ndash124 1994

[3] M A Chaudhry A Qadir M Rafique and S M ZubairldquoExtension of Eulerrsquos beta functionrdquo Journal of Computationaland Applied Mathematics vol 78 no 1 pp 19ndash32 1997

12 Journal of Mathematics

[4] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions John Wiley amp Sons New York NY USA 1984

[5] M E H Ismail and J Pitman ldquoAlgebraic evaluations of someEuler integrals duplication formulae for Appellrsquos hypergeomet-ric function 119865

1 and Brownian variationsrdquo Canadian Journal of

Mathematics vol 52 no 5 pp 961ndash981 2000[6] J Pitman ldquoBrownian motion bridge excursion and meander

characterized by sampling at independent uniform timesrdquoElectronic Journal of Probability vol 11 no 4 pp 1ndash33 1999

[7] Y A Brychkov Handbook of Special Functions DerivativesIntegrals Series and Other Formulas CRC Press New York NYUSA 2008

[8] G Dattoli S Lorenzutta G Maino A Torre and C CesaranoldquoGeneralized Hermite polynomials and super-Gaussian formsrdquoJournal of Mathematical Analysis and Applications vol 203 no3 pp 597ndash609 1996

[9] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[10] P Appell and J K de Feriet Fonctions Hypergeometriques etHyperspheriques Polynomes drsquoHermite Gauthier-Villars ParisFarnce 1926

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962

[12] M Lahiri ldquoOn a generalisation of Hermite polynomialsrdquoProceedings of the American Mathematical Society vol 27 pp117ndash121 1971

[13] G Dattoli and A Torre ldquoOperatorial methods and two variableLaguerre polynomialsrdquo Atti della Accademia delle Scienze diTorino Classe di Scienze Fisiche Matematiche e Naturali vol132 pp 1ndash7 1998

[14] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009

[15] S RomanTheUmbral Calculus Academic Press NewYorkNYUSA 1984

[16] S Khan M W Al-Saad and R Khan ldquoLaguerre-based Appellpolynomials properties and applicationsrdquo Mathematical andComputer Modelling vol 52 no 1-2 pp 247ndash259 2010

[17] S Khan M W Al-Saad and G Yasmin ldquoSome properties ofHermite-based Sheffer polynomialsrdquo Applied Mathematics andComputation vol 217 no 5 pp 2169ndash2183 2010

[18] E D Rainville Special Functions Macmillan New York NYUSA 1960

[19] E D Rainville SpecialFunctions Chelsea Publishing NewYorkNY USA 1971

[20] B C Berndt Ramanujanrsquos NotebooksmdashPart I Springer NewYork NY USA 1985

[21] G H Hardy Ramanujan Twelve Lectures on Subjects Suggestedby His Life and Work Cambridge University Press CambridgeUK 1940

[22] D Babusci G Dattoli G H E Duchamp K Gorska and K APenson ldquoDefinite integrals and operational methodsrdquo AppliedMathematics and Computation vol 219 no 6 pp 3017ndash30212012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Mathematics

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119867119904119899(1199091 1199092 1199093)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(63)

Taking 1199092= 119909

3= 0 and replacing 119909

1by 119909 in (63) and

using the relation119867119904119899(119909 0 0) = 119904

119899(119909) [17 page 16 (23)] we

get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

119860 (119905(119906 minus 119886)120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)+ 119909119867 (119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119904119899(119909)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899 119887)119905119899

119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(64)

where 119904119899(119909) are the Sheffer polynomials [15 18 19]

4 Concluding Remarks

In this paper we have derived the evaluations of certain Eulertype integralsWehave also established a theoremand appliedit to obtain evaluations of some integrals in terms of EBF119861(120572 120573 119887) We remark that these results can be extended tomultivariable case

To give an example we consider the following integralrepresentation [5 page 965 (20)] of Lauricellarsquos multiplehypergeometric series 119865(119899)

119863[4]

int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

119889119905

= 119861 (120572 120573) 119865(119899)

119863[120572 120572

1 1205722 120572

119899 120572 + 120573 119909

1 1199092 119909

119899]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(65)

Now taking 120601(119905) = prod119899

119894=1(1 minus 119909

119894119905)minus120572119894 120595(119905) = 1119905(1 minus 119905)

119886 = 0 and 119890 = 120574 = 1 in (15) using integral (65) in the rhs

after expanding119903119865119905[

(119891119903)

119889120595(119905)

(119892119905)

] in series and simplifying weget

119868119889120572120573101

[

119899

prod

119894=1

(1 minus 119909119894119905)minus120572119894

1

119905 (1 minus 119905)]

=

infin

sum

11989811198982119898119899=0

(120572)1198981+1198982+sdotsdotsdot+119898

119899

(1205721)1198981

(1205722)1198982

sdot sdot sdot (120572119899)119898119899

(120572 + 120573)1198981+1198982+sdotsdotsdot+119898

119899

1199091198981

1

1198981

1199091198982

2

1198982sdot sdot sdot

119909119898119899

119899

119898119899

times119903+2119865119905+2

[[[[[[[

[

(119891119903) 1

2(1 minus 120572 minus 120573 minus

119899

sum

119894=1

119898119894) 1 minus

1

2(120572 + 120573 +

119899

sum

119894=1

119898119894)

4119889

(119892119905) 1 minus (120572 +

119899

sum

119894=1

119898119894) 1 minus 120573

]]]]]]]

]

(Re (120572) Re (120573) gt 0max 100381610038161003816100381611990911003816100381610038161003816 10038161003816100381610038161199092

1003816100381610038161003816 1003816100381610038161003816119909119899

1003816100381610038161003816 lt 1)

(66)

We note that for 1205723= 120572

4= sdot sdot sdot = 0 (66) reduces to (17)

Further we extendTheorem 1 as follows

Theorem 5 Let the conditions for 119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

)

times exp(minus 119887(119888 minus 119886)2

(119906 minus 119886) (119888 minus 119906)) 119889119906

=

infin

sum

11989911989811198982119898119904=0

(119888 minus 119886)120583+(120588+120590)119899+119898

1+1198982+sdotsdotsdot+119898

119904minus1

times 119888119899119892119899(1199091 1199092 119909

119903)

times 119861(120582 + 120588119899 +

119904

sum

119894=1

119898119894 120583 minus 120582 + 120590119899 119887)

119904

prod

119894=1

(120572119894)119898119894

119910119898119894

119894

119898119894

119905119899

(Re (120583) gt Re (120582) gt 0Re (119887) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(67)

Journal of Mathematics 11

The proof of this theorem is similar to that of Theorem 1The following are the consequences of Theorem 5

Corollary 6 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and119904 = 2 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

2

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899)

times 1198651[120582 + 120588119899 120572

1 1205722 120583 + (120588 + 120590) 119899

(119888 minus 119886) 1199101 (119888 minus 119886) 119910

2] 119905119899

(Re (120583) gt Re (120582) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(68)

Corollary 7 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and120588 = 120590 = 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905 (119906 minus 119886) (119888 minus 119906)) 119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904] 119905119899

(Re (120583) gt Re (120582) gt 0)

(69)

Example 8 Applying Corollary 7 to generating function (37)of the multivariable Hermite polynomials 119867(12119903)

119899(1199091 1199092

119909119903) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times exp(119903

sum

119895=1

119909119895(119905 (119906 minus 119886) (119888 minus 119906))

119895

)119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904]119905119899

119899

(Re (120583) gt Re (120582) gt 0)

(70)Remark 9 Taking 119886 = 0 and 119888 = 1 in (67) we get an extensionof integral (34)

Remark 10 Taking 119886 = minus1 and 119888 = 1 in (67) we get anextension of integral (35)

Clearly it appears that by applying Theorem 5 andCorollaries 6 and 7 to the generating functions defined by(37) (44) (45) (54) (57) (58) and (62) and also to someother generating functions a number of interesting integralscan be obtained

Further it is remarked that the operational methodsprovide useful tools to estimate specific theorems in variousfields of analysisThis approach requires the validation of anyof its consequences by a more rigorous procedure This isindeed the case of the Ramanujan master theorem [20 21]An operational method already employed to formulate ageneralization of the Ramanujan master theorem is appliedfor the evaluation of certain integrals by Babusci et al [22]

The technique adopted in [22] provides a very flexible andpowerful tool yielding new results encompassing differentaspects of the theory of special functions The possibility ofusing the method outlined in [22] for the integrals evaluatedin this article is a further research problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to thereviewer(s) for helpful suggestions and comments towardsthe improvement of this paper

References

[1] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985

[2] M A Chaudhry and S M Zubair ldquoGeneralized incompletegamma functions with applicationsrdquo Journal of Computationaland Applied Mathematics vol 55 no 1 pp 99ndash124 1994

[3] M A Chaudhry A Qadir M Rafique and S M ZubairldquoExtension of Eulerrsquos beta functionrdquo Journal of Computationaland Applied Mathematics vol 78 no 1 pp 19ndash32 1997

12 Journal of Mathematics

[4] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions John Wiley amp Sons New York NY USA 1984

[5] M E H Ismail and J Pitman ldquoAlgebraic evaluations of someEuler integrals duplication formulae for Appellrsquos hypergeomet-ric function 119865

1 and Brownian variationsrdquo Canadian Journal of

Mathematics vol 52 no 5 pp 961ndash981 2000[6] J Pitman ldquoBrownian motion bridge excursion and meander

characterized by sampling at independent uniform timesrdquoElectronic Journal of Probability vol 11 no 4 pp 1ndash33 1999

[7] Y A Brychkov Handbook of Special Functions DerivativesIntegrals Series and Other Formulas CRC Press New York NYUSA 2008

[8] G Dattoli S Lorenzutta G Maino A Torre and C CesaranoldquoGeneralized Hermite polynomials and super-Gaussian formsrdquoJournal of Mathematical Analysis and Applications vol 203 no3 pp 597ndash609 1996

[9] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[10] P Appell and J K de Feriet Fonctions Hypergeometriques etHyperspheriques Polynomes drsquoHermite Gauthier-Villars ParisFarnce 1926

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962

[12] M Lahiri ldquoOn a generalisation of Hermite polynomialsrdquoProceedings of the American Mathematical Society vol 27 pp117ndash121 1971

[13] G Dattoli and A Torre ldquoOperatorial methods and two variableLaguerre polynomialsrdquo Atti della Accademia delle Scienze diTorino Classe di Scienze Fisiche Matematiche e Naturali vol132 pp 1ndash7 1998

[14] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009

[15] S RomanTheUmbral Calculus Academic Press NewYorkNYUSA 1984

[16] S Khan M W Al-Saad and R Khan ldquoLaguerre-based Appellpolynomials properties and applicationsrdquo Mathematical andComputer Modelling vol 52 no 1-2 pp 247ndash259 2010

[17] S Khan M W Al-Saad and G Yasmin ldquoSome properties ofHermite-based Sheffer polynomialsrdquo Applied Mathematics andComputation vol 217 no 5 pp 2169ndash2183 2010

[18] E D Rainville Special Functions Macmillan New York NYUSA 1960

[19] E D Rainville SpecialFunctions Chelsea Publishing NewYorkNY USA 1971

[20] B C Berndt Ramanujanrsquos NotebooksmdashPart I Springer NewYork NY USA 1985

[21] G H Hardy Ramanujan Twelve Lectures on Subjects Suggestedby His Life and Work Cambridge University Press CambridgeUK 1940

[22] D Babusci G Dattoli G H E Duchamp K Gorska and K APenson ldquoDefinite integrals and operational methodsrdquo AppliedMathematics and Computation vol 219 no 6 pp 3017ndash30212012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 11

The proof of this theorem is similar to that of Theorem 1The following are the consequences of Theorem 5

Corollary 6 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and119904 = 2 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

2

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905(119906 minus 119886)

120588

(119888 minus 119906)120590

) 119889119906

=

infin

sum

119899=0

(119888 minus 119886)120583+(120588+120590)119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times 119861 (120582 + 120588119899 120583 minus 120582 + 120590119899)

times 1198651[120582 + 120588119899 120572

1 1205722 120583 + (120588 + 120590) 119899

(119888 minus 119886) 1199101 (119888 minus 119886) 119910

2] 119905119899

(Re (120583) gt Re (120582) gt 0 120588 120590 ge 0 120588 + 120590 gt 0)

(68)

Corollary 7 Let the conditions for119866(1199091 1199092 119909

119903 119905) defined

by (31) be the same as in Theorem 1 and then for 119887 = 0 and120588 = 120590 = 1 one has

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times 119866 (1199091 1199092 119909

119903 119905 (119906 minus 119886) (119888 minus 119906)) 119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119888119899119892119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904] 119905119899

(Re (120583) gt Re (120582) gt 0)

(69)

Example 8 Applying Corollary 7 to generating function (37)of the multivariable Hermite polynomials 119867(12119903)

119899(1199091 1199092

119909119903) we get

int

119888

119886

(119906 minus 119886)120582minus1

(119888 minus 119906)120583minus120582minus1

times

119904

prod

119894=1

(1 minus 119910119894(119906 minus 119886))

minus120572119894

times exp(119903

sum

119895=1

119909119895(119905 (119906 minus 119886) (119888 minus 119906))

119895

)119889119906

= 119861 (120582 120583 minus 120582)

infin

sum

119899=0

(119888 minus 119886)120583+2119899minus1

119867(12119903)

119899(1199091 1199092 119909

119903)

times(120582)

119899(120583 minus 120582)

119899

(120583)2119899

times 119865(119904)

119863[120582 + 119899 120572

1 1205722 120572

119904 120583 + 2119899 (119888 minus 119886) 119910

1

(119888 minus 119886) 1199102 (119888 minus 119886) 119910

119904]119905119899

119899

(Re (120583) gt Re (120582) gt 0)

(70)Remark 9 Taking 119886 = 0 and 119888 = 1 in (67) we get an extensionof integral (34)

Remark 10 Taking 119886 = minus1 and 119888 = 1 in (67) we get anextension of integral (35)

Clearly it appears that by applying Theorem 5 andCorollaries 6 and 7 to the generating functions defined by(37) (44) (45) (54) (57) (58) and (62) and also to someother generating functions a number of interesting integralscan be obtained

Further it is remarked that the operational methodsprovide useful tools to estimate specific theorems in variousfields of analysisThis approach requires the validation of anyof its consequences by a more rigorous procedure This isindeed the case of the Ramanujan master theorem [20 21]An operational method already employed to formulate ageneralization of the Ramanujan master theorem is appliedfor the evaluation of certain integrals by Babusci et al [22]

The technique adopted in [22] provides a very flexible andpowerful tool yielding new results encompassing differentaspects of the theory of special functions The possibility ofusing the method outlined in [22] for the integrals evaluatedin this article is a further research problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to thereviewer(s) for helpful suggestions and comments towardsthe improvement of this paper

References

[1] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985

[2] M A Chaudhry and S M Zubair ldquoGeneralized incompletegamma functions with applicationsrdquo Journal of Computationaland Applied Mathematics vol 55 no 1 pp 99ndash124 1994

[3] M A Chaudhry A Qadir M Rafique and S M ZubairldquoExtension of Eulerrsquos beta functionrdquo Journal of Computationaland Applied Mathematics vol 78 no 1 pp 19ndash32 1997

12 Journal of Mathematics

[4] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions John Wiley amp Sons New York NY USA 1984

[5] M E H Ismail and J Pitman ldquoAlgebraic evaluations of someEuler integrals duplication formulae for Appellrsquos hypergeomet-ric function 119865

1 and Brownian variationsrdquo Canadian Journal of

Mathematics vol 52 no 5 pp 961ndash981 2000[6] J Pitman ldquoBrownian motion bridge excursion and meander

characterized by sampling at independent uniform timesrdquoElectronic Journal of Probability vol 11 no 4 pp 1ndash33 1999

[7] Y A Brychkov Handbook of Special Functions DerivativesIntegrals Series and Other Formulas CRC Press New York NYUSA 2008

[8] G Dattoli S Lorenzutta G Maino A Torre and C CesaranoldquoGeneralized Hermite polynomials and super-Gaussian formsrdquoJournal of Mathematical Analysis and Applications vol 203 no3 pp 597ndash609 1996

[9] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[10] P Appell and J K de Feriet Fonctions Hypergeometriques etHyperspheriques Polynomes drsquoHermite Gauthier-Villars ParisFarnce 1926

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962

[12] M Lahiri ldquoOn a generalisation of Hermite polynomialsrdquoProceedings of the American Mathematical Society vol 27 pp117ndash121 1971

[13] G Dattoli and A Torre ldquoOperatorial methods and two variableLaguerre polynomialsrdquo Atti della Accademia delle Scienze diTorino Classe di Scienze Fisiche Matematiche e Naturali vol132 pp 1ndash7 1998

[14] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009

[15] S RomanTheUmbral Calculus Academic Press NewYorkNYUSA 1984

[16] S Khan M W Al-Saad and R Khan ldquoLaguerre-based Appellpolynomials properties and applicationsrdquo Mathematical andComputer Modelling vol 52 no 1-2 pp 247ndash259 2010

[17] S Khan M W Al-Saad and G Yasmin ldquoSome properties ofHermite-based Sheffer polynomialsrdquo Applied Mathematics andComputation vol 217 no 5 pp 2169ndash2183 2010

[18] E D Rainville Special Functions Macmillan New York NYUSA 1960

[19] E D Rainville SpecialFunctions Chelsea Publishing NewYorkNY USA 1971

[20] B C Berndt Ramanujanrsquos NotebooksmdashPart I Springer NewYork NY USA 1985

[21] G H Hardy Ramanujan Twelve Lectures on Subjects Suggestedby His Life and Work Cambridge University Press CambridgeUK 1940

[22] D Babusci G Dattoli G H E Duchamp K Gorska and K APenson ldquoDefinite integrals and operational methodsrdquo AppliedMathematics and Computation vol 219 no 6 pp 3017ndash30212012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Journal of Mathematics

[4] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions John Wiley amp Sons New York NY USA 1984

[5] M E H Ismail and J Pitman ldquoAlgebraic evaluations of someEuler integrals duplication formulae for Appellrsquos hypergeomet-ric function 119865

1 and Brownian variationsrdquo Canadian Journal of

Mathematics vol 52 no 5 pp 961ndash981 2000[6] J Pitman ldquoBrownian motion bridge excursion and meander

characterized by sampling at independent uniform timesrdquoElectronic Journal of Probability vol 11 no 4 pp 1ndash33 1999

[7] Y A Brychkov Handbook of Special Functions DerivativesIntegrals Series and Other Formulas CRC Press New York NYUSA 2008

[8] G Dattoli S Lorenzutta G Maino A Torre and C CesaranoldquoGeneralized Hermite polynomials and super-Gaussian formsrdquoJournal of Mathematical Analysis and Applications vol 203 no3 pp 597ndash609 1996

[9] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[10] P Appell and J K de Feriet Fonctions Hypergeometriques etHyperspheriques Polynomes drsquoHermite Gauthier-Villars ParisFarnce 1926

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962

[12] M Lahiri ldquoOn a generalisation of Hermite polynomialsrdquoProceedings of the American Mathematical Society vol 27 pp117ndash121 1971

[13] G Dattoli and A Torre ldquoOperatorial methods and two variableLaguerre polynomialsrdquo Atti della Accademia delle Scienze diTorino Classe di Scienze Fisiche Matematiche e Naturali vol132 pp 1ndash7 1998

[14] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009

[15] S RomanTheUmbral Calculus Academic Press NewYorkNYUSA 1984

[16] S Khan M W Al-Saad and R Khan ldquoLaguerre-based Appellpolynomials properties and applicationsrdquo Mathematical andComputer Modelling vol 52 no 1-2 pp 247ndash259 2010

[17] S Khan M W Al-Saad and G Yasmin ldquoSome properties ofHermite-based Sheffer polynomialsrdquo Applied Mathematics andComputation vol 217 no 5 pp 2169ndash2183 2010

[18] E D Rainville Special Functions Macmillan New York NYUSA 1960

[19] E D Rainville SpecialFunctions Chelsea Publishing NewYorkNY USA 1971

[20] B C Berndt Ramanujanrsquos NotebooksmdashPart I Springer NewYork NY USA 1985

[21] G H Hardy Ramanujan Twelve Lectures on Subjects Suggestedby His Life and Work Cambridge University Press CambridgeUK 1940

[22] D Babusci G Dattoli G H E Duchamp K Gorska and K APenson ldquoDefinite integrals and operational methodsrdquo AppliedMathematics and Computation vol 219 no 6 pp 3017ndash30212012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of