Research ArticleFinite Integral Formulas Involving MultivariableAleph-Functions

Hagos Tadesse D L Suthar andMinilik Ayalew

Department of Mathematics Wollo University Dessie PO Box 1145 Amhara Region Ethiopia

Correspondence should be addressed to D L Suthar dlsuthargmailcom

Received 6 March 2019 Revised 10 July 2019 Accepted 10 July 2019 Published 21 August 2019

Academic Editor Kai Diethelm

Copyright copy 2019 Hagos Tadesse et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The integrals evaluated are the products of multivariable Aleph-functions with algebraic functions Jacobi polynomials Legendrefunctions Bessel-Maitland functions and general class of polynomials The main results of our paper are quite general in natureand competent at yielding a very large number of integrals involving polynomials and various special functions occurring in theproblem of mathematical analysis and mathematical physics

1 Introduction and Preliminaries

Throughout this paper consider CRR+Zminus0 and N to be a

set of complex numbers positive real numbers nonpositiveintegers and positive integers respectivelyThemultivariableAleph (alefsym) function of several complex variables generalizesthe multivariable I-function recently studied by Sharma andAhmad [1] which itself is a generalization of G- and H-functions of multiple variables as

alefsym(1199111 1199112 119911119903)

= alefsym01198991198981119899111989821198992 119898119903 119899119903119901119894119902119894 120591119894119877119901119894(1) 119902119894(1) 120591119894(1) 119877

(1) 119901119894(119903)

119902119894(119903)

120591119894(119903)

119877(119903)

1199111119911119903

| 119860 119861119862 119863

= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 sdot sdot sdot 120585119903) 119903prod

119894=1

120601119894 (120585119894) 119911120585119894119894 1198891205851 sdot sdot sdot 119889120585119903(1)

where 120596 = radicminus1

119860 = [(119886119895 120572(1)119895 sdot sdot sdot 120572(119903)119895 )1119899

] [120591119894 (119886119895119894 120572(1)119895119894 sdot sdot sdot 120572(119903)119895119894 )119899+1119901119894]119861 = [(119888(1)119895 120574(1)119895 )

11198991] [120591119894(1) (119888(1)119895119894(1)

120574(1)119895119894(1)

)1198991+1119901

(1)119894

] sdot sdot sdot [(119888(119903)119895 120574(119903)119895 )1119899119903

] [120591119894(119903) (119888(119903)119895119894(1) 120574(119903)

119895119894(1))119899119903+1119901

(119903)119894

]119862 = [ 120591119894 (119887119895119894 120573(1)

119895119894 sdot sdot sdot 120573(119903)119895119894 )119898+1119902119894

]119863 = [(119889(1)119895 120575(1)119895 )

11198981] [120591119894(1) (119889(1)119895119894(1)

120575(1)119895119894(1)

)1198981+1119902

(1)119894

] sdot sdot sdot [(119889(119903)119895 120575(119903)119895 )1119898119903

] [120591119894(119903) (119889(119903)119895119894(119903) 120574(119903)

119895119894(119903))119898119903+1119902

(119903)119894

]

(2)

120595 (1205851 sdot sdot sdot 120585119903) = prod119899119895=1Γ (1 minus 119886119895 + sum119903

119896=1 120572(119896)119895 120585119896)sum119877

119894=1 [120591119894prod119901119894119895=119899+1Γ (119886119895119894 minus sum119903

119896=1 120572(119896)119895119894 120585119896)prod119902119894119895=1Γ (1 minus 119887119895119894 + sum119903

119896=1 120573(119896)119895119894 120585119896)] (3)

HindawiJournal of Applied MathematicsVolume 2019 Article ID 6821797 10 pageshttpsdoiorg10115520196821797

2 Journal of Applied Mathematics

120601119896 (120585119896) = prod119898119896119895=1Γ (119889(119896)119895 minus 120575(119896)119895 120585119896)prod119899119896

119895=1Γ (1 minus 119888(119896)119895 + 120574(119896)119895 120585119896)sum119877(119896)

119894(119896)=1 [120591119894(119896)prod119902119894(119896)

119895=119898119896+1Γ (1 minus 119889(119896)

119895119894(119896)+ 120575(119896)

119895119894(119896)120585119896)prod119901

119894(119896)

119895=119899119896+1Γ (119888(119896)

119895119894(119896)minus 120574(119896)

119895119894(119896)120585119896)] (4)

and 119886119895(119895 = 1 sdot sdot sdot 119901) 119887119895(119895 = 1 sdot sdot sdot 119902) 119888(119896)119895 (119895 = 1 sdot sdot sdot 119899119896)119888(119896)119895119894(119896)

(119895 = 119899119896 + 1 sdot sdot sdot 119901119894(119896)) 119889(119896)119895 (119895 = 1 sdot sdot sdot 119898119896) 119889(119896)119895119894(119896)

(119895 = 119898119896 +1 sdot sdot sdot 119902119894(119896)) (119896 = 1 sdot sdot sdot 119903 119894 = 1 sdot sdot sdot 119877 and 119894(119896) = 1 sdot sdot sdot 119877(119896)

are complex numbers

119880(119896)119894 = 119899sum

119895=1

120572(119896)119895 + 120591119894119901119894sum

119895=119899+1

120572(119896)119895119894 + 119899119896sum119895=1

120574(119896)119895 + 120591119894(119896)119901119894(119896)sum119895=119899119896+1

120574(119896)119895119894(119896)

minus 120591119894119902119894sum119895=1

120573(119896)

119895119894(119896)minus 119898119896sum

119895=1

120575(119896)119895 minus 120591119894(119896)119902119894(119896)sum

119895=119898119896+1

120575(119896)119895119894(119896)

le 0(5)

120591119894(119894 = 1 sdot sdot sdot 119877) 120591119894(119896)(119894(119896) = 1 sdot sdot sdot 119877(119896)) are positive realnumbers

The integration path 119871120596120574infin (120574 isin 119877) extends from 120574 minus 120596infinto 120574 + 120596infin and the poles of Γ(119889(119896)119895 minus 120575(119896)119895 120585119896) 119895 = 1 119898119896do not coincide with the poles of Γ(1 minus 119886119895 + sum119903

119894=1 120572(119896)119895 120585119896) 119895 =1 119899 and Γ(1 minus 119888(119896)119895 + 120574(119896)119895 120585119896) 119895 = 1 119899119896 to the left of thecontour 119871119896

The existence condition for multiple Mellin-Barnes typecontours (1) can be given below

1003816100381610038161003816arg 1199111198961003816100381610038161003816 lt 12119860(119896)119894 120587 (6)

where

119860(119896)119894 = 119899sum

119895=1

120572(119896)119895 minus 120591119894119901119894sum

119895=119899+1

120572(119896)119895119894 minus 120591119894119902119894sum119895=1

120573(119896)119895119894 + 119899119896sum

119895=1

120574(119896)119895

minus 120591119894(119896)119901119894(119896)sum

119895=119899119896+1

120574(119896)119895119894(119896)

+ 119898119896sum119895=1

120575(119896)119895 minus 120591119894(119896)119902119894(119896)sum

119895=119898119896+1

120575(119896)119895119894(119896)

gt 0(7)

with = 1 sdot sdot sdot 119903 119894 = 1 sdot sdot sdot 119877 and 119894(119896) = 1 sdot sdot sdot 119877(119896)Remark 1 By setting 120591119894 = 120591119894(119896) = 1 the multivariable Aleph-function reduces to multivariable I-function (see [1ndash3]])

Remark 2 By setting 120591119894 = 120591119894(119896) = 1 (119896 isin 1 119903) and119877 = 119877(1) = = 119877(119903) = 1 the multivariable Aleph-functionreduces to multivariable H-function defined by Srivastava etal [4]

Remark 3 When we set 119903 = 1 the multivariable Aleph-function reduces to Aleph-function of one variable definedby Sdland [5]

For the definition of the H- functionalefsym-function and itsmore generalization the interested reader may refer to thepapers [6ndash13]

From Rainville [14] the integral representation of thegamma function Γ(119909) is defined as

Γ (119909) = intinfin

0119890minus119905119905119909minus1119889119905 R (119909) gt 0 (8)

And also the beta integral is defined as follows

119861 (119909 119910) = int1

0119905119909minus1 (1 minus 119905)119910minus1 119889119905 = Γ (119909) Γ (119910)

Γ (119909 + 119910)= 119861 (119910 119909) (R (119909) R (119910) gt 0)

(9)

Further We will use the following notations in this paper

119880 = 1198981 1198991 1198982 1198992 119898119903 119899119903119881 = 119901119894(1) 119902119894(1) 120591119894(1) 119877(1)119882 = 119901119894(119903) 119902119894(119903) 120591119894(119903) 119877(119903)

(10)

2 Integrals Involving Multivariable Aleph-Function with Algebraic Function

In this section we evaluate integrals the product of multi-variable Aleph-functions with various algebraic functions

1198681 = int1

0119909minus120588 (1 minus 119909)120588minus120590minus1 alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205961 119911119903119909120596119903] 119889119909

= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

01199091minus120588+sum119903119894=1 120596119894120585119894minus1 (1 minus 119909)120588minus120590minus1 1198891199091198891205851 119889120585119903

= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894)

sdot 119911120585119894119894 119861(1 minus 120588 + 119903sum119894=1

120596119894120585119894 120588 minus 120590)1198891205851 sdot sdot sdot 119889120585119903 = Γ (120588 minus 120590)sdot 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903)sdot 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 Γ (1 minus 120588 + sum119903119894=1 120596119894120585119894)Γ (1 minus 120590 + sum119903119894=1 120596119894120585119894)1198891205851 sdot sdot sdot 119889120585119903 = Γ (120588 minus 120590)

sdot alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

1199111119911119903

| (120588 1205961 120596119903) 119860 119861(120590 1205961 120596119903) 119862 119863

]]]]]

(11)

Journal of Applied Mathematics 3

1198682 = int1

0119909120588minus1 (1 minus 119909)120590minus1 alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205961 119911119903119909120596119903] 119889119909

= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894sdot int1

0119909120588+sum119903119894=1 120596119894120585119894minus1 (1 minus 119909)120590minus1 1198891199091198891205851 119889120585119903 = 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 119861(120588 + 119903sum119894=1

120596119894120585119894 120590)1198891205851 sdot sdot sdot 119889120585119903= Γ (120590) 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) 119911120585119894119894sdot Γ (120588 + sum119903

119894=1 120585119894)Γ (120588 + 120590 + sum119903119894=1 120585119894)1198891205851 sdot sdot sdot 119889120585119903 = Γ (120590)

sdot alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

1199111119911119903

| (1 minus 120588 1205961 120596119903) 119860 119861(1 minus 120588 minus 120590 1205961 120596119903) 119862 119863

]]]]]

(12)

1198683 = intinfin

1119909minus120588 (119909 minus 1)120590minus1 alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205961 119911119903119909120596119903] 119889119909

= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894sdot intinfin

1119909minus120588+sum119903119894=1 120596119894120585119894 (119909 minus 1)120590minus1 1198891199091198891205851 119889120585119903

(13)

Putting 119909 = 119905 + 1 we have 119889119909 = 119889119905 and we use the followingrelation

Γ (120572) Γ (120573)Γ (120572 + 120573) = intinfin

0119909120572minus1 (119909 + 1)minus(120572+120573) 119889119909 = intinfin

0119909120573minus1 (119909

+ 1)minus(120572+120573) 119889119909(14)

1198683 = 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894sdot intinfin

0119905120590minus1 (1 + 119905)minus(120588minussum119903119894=1 120596119894120585119894) 119889119905 1198891205851 119889120585119903 = Γ (120590)

sdot 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903)sdot 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 Γ (120588 minus 120590 minus sum119903119894=1 120596119894120585119894)Γ (120588 minus sum119903

119894=1 120596119894120585119894) 1198891205851 sdot sdot sdot 119889120585119903 = Γ (120590)

sdot alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

1199111119911119903

| (120588 1205961 120596119903) 119860 119861(120588 minus 120590 1205961 120596119903) 119862 119863

]]]]]

(15)

1198684 = intinfin

0119909120588minus1 (119909 + 120573)minus120590 alefsym0119899119880

119901119894119902119894120591119894119877119881119882[11991111199091205961 119911119903119909120596119903] 119889119909

= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894sdot intinfin

0119909120588+sum119903119894=1 120596119894120585119894minus1 (119909 + 120573)minus120590 1198891199091198891205851 119889120585119903

(16)

Setting 119909 = 120573119905 implies 119889119909 = 120573119889119905 and then we obtain thefollowing

= 120573120588minus120590

(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (120573120596119894119911119894)120585119894

sdot intinfin

0119905120588+sum119903119894=1 120596119894120585119894minus1 (1 + 119905)minus120590 119889119905 1198891205851 119889120585119903

(17)

By applying (14) we have the following

= 120573120588minus120590

Γ (120590) 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (120573120596119894119911119894)120585119894

times Γ(120588 + 119903sum119894=1

120596119894120585119894)Γ(120590 minus 120588 minus 119903sum119894=1

120596119894120585119894)1198891205851 119889120585119903(18)

= 120573120588minus120590

Γ (120590)

sdot alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

12057312059611199111120573120596119903119911119903

| (1 minus 120588 1205961 120596119903) 119860 119861(120590 minus 120588 1205961 120596119903) 119862 119863

]]]]](19)

1198685 = int1

minus1(1 minus 119909)120588 (1 + 119909)120590

sdot alefsym0119899119880119901119894119902119894 120591119894119877119881119882

[1199111 (1 minus 119909)1205831 119911119903 (1 minus 119909)120583119903] 119889119909= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) 119911120585119894119894sdot int1

minus1(1 minus 119909)120588+sum119903119894=1 120583119894120585119894 (1 + 119909)120590 1198891199091198891205851 119889120585119903

(20)

Now we use the following formula ([14] p261)

int1

minus1(1 minus 119909)119899+120572 (1 + 119909)119899+120573 119889119909= 22119899+120572+120573+1119861 (1 + 120572 + 119899 1 + 120573 + 119899)

(21)

Hence we arrive at

= 2120588+120590+1Γ (1 + 120590) 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (2120583119894119911119894)120585119894 times Γ (1 + 120588 + sum119903119894=1 120583119894120585119894)Γ (2 + 120588 + 120590 + 120583sum119903

119894=1 120583119894120585119894)1198891205851 119889120585119903 (22)

= 2120588+120590+1Γ (1 + 120590)alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

2120583111991112120583119903119911119903

| (minus120588 1205831 120583119903) 119860 119861(minus1 minus 120590 minus 120588 1205831 120583119903) 119862 119863

]]]]] (23)

4 Journal of Applied Mathematics

3 Integrals Involving Multivariable Aleph-Function with Jacobi Polynomials

The Jacobi polynomial ([15] 4212)) with parameters 120572 120573 isinR is defined by

119901(120572120573)119899 (119909)= (1 + 120572)119899119899 21198651 [minus119899 1 + 120572 + 120573 + 119899 1 + 120572 1 minus 1199092 ]

119899 ge 1(24)

where 21198651[] is the classical hypergeometric function Bysubstituting 120572 = 120573 = 0 the Jacobi polynomial (24) reducesto Lagrange polynomial ( [14] p 157) as

119901(120572120573)119899 (1) = (1 + 120572)119899119899 (25)

In this section we derive integral formulas involving multi-variable Aleph-functions multiplied by Jacobi polynomials

1198686= int1

minus1119909120582 (1

minus 119909)120572 (1 + 119909)120583 119875(120572120573)119899 (119909)alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[1199111 (1 + 119909)1198971

119911119903 (1 + 119909)119897119903] 119889119909

= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894)

sdot 119911120585119894119894 times int1

minus1119909120582 (1 minus 119909)120572 (1 + 119909)120583+sum119903119894=1 119897119894120585119894

sdot 119875(120572120573)119899 (119909) 119889119909 1198891205851 119889120585119903

(26)

Next we use the following formula

int1

minus1119909120582 (1 minus 119909)120572 (1 + 119909)120583 119901(120572120573)

119899 (119909) 119889119909= (minus1)119899 2120572+120583+1Γ (120583 + 1) Γ (119899 + 120572 + 1) Γ (120583 + 120573 + 1)

119899Γ (120583 + 120573 + 119899 + 1) Γ (120583 + 120572 + 119899 + 2)times 31198652 [ minus120582 120583 + 120573 + 1 120583 + 1

120583 + 120573 + 119899 + 1 120583 + 120572 + 119899 + 2 1] (27)

where 120572 gt minus1 and 120573 gt minus1 Also 31198652 is the special case ofgeneralized hypergeometric series

Then we have the following

= 1(2120587120596)119903sdot int

1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) rprod119894=1

120601119894 (120585119894) 119911120585119894119894 (minus1)119899 2120572+120583+sum119903119894=1 119897119894120585119894+1119899times Γ (120583 + sum119903

119894=1 119897119894120585119894 + 1) Γ (119899 + 120572 + 1) Γ (120583 + sum119903119894=1 119897119894120585119894 + 120573 + 1)

Γ (120583 + sum119903119894=1 119897119894120585119894 + 120573 + 119899 + 1) Γ (120583 + sum119903

119894=1 119897119894120585119894 + 120572 + 119899 + 2)

times 31198652

[[[[

minus120582 120583 + 119903sum119894=1

119897119894120585119894 + 120573 + 1 120583 + 119903sum119894=1

119897119894120585119894 + 1120583 + 119903sum

119894=1119897119894120585119894 + 120573 + 119899 + 1 120583 + 119903sum

119894=1119897119894120585119894 + 120572 + 119899 + 2 1]]]

]1198891205851

119889120585119903

(28)

Using the definition of hypergeometric function and somesimplifications the above expression becomes

1198686 = (minus1)119899 2120572+120583+1Γ (120572 + 119899 + 1)119899infinsum119896=0

(minus120582)119896 1119896119896 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) times (2119897119894119911119894)120585119894 Γ (120583 + sum119903119894=1 119897119894120585119894 + 120573 + 119896 + 1) Γ (120583 + sum119903

119894=1 119897119894120585119894 + 119896 + 1)Γ (120583 + sum119903

119894=1 119897119894120585119894 + 120573 + 119899 + 119896 + 1) Γ (120583 + sum119903119894=1 119897119894120585119894 + 120572 + 119899 + 119896 + 2)1198891205851

119889120585119903 = (minus1)119899 2120572+120583+1Γ (120572 + 119899 + 1)119899infinsum119896=0

(minus120582)119896 1119896119896

sdot alefsym0119899+2119880119901119894+2119902119894+2120591119894 119877119881119882

[[[[[

2119897111991112119897119903119911119903

| (minus120583 minus 120573 minus 119896 1198971 119897119903) (minus120583 minus 119896 1198971 119897119903) 119860 119861(minus120583 minus 120573 minus 119899 minus 119896 1198971 119897119903) (minus1 minus 120583 minus 120572 minus 119899 minus 119896 1198971 119897119903) 119862 119863

]]]]]

(29)

Journal of Applied Mathematics 5

Thus 120572 gt minus1 120573 gt minus1R(120582) gt minus1 and |arg 119911| lt (12)120587Ω

1198687= int1

minus1(1

minus 119909)120575 (1 + 119909)V 119875(120583V)119899 (119909) 119875(120588120590)

119898 (119909) times alefsym0119899119880119901119894119902119894 120591119894119877119881119882

[1199111 (1minus 119909)1198971 119911119903 (1 minus 119909)119897119903] 119889119909= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) 119911120585119894119894 times int1

minus1(1

minus 119909)120575+sum119903119894=1 119897119894120585119894 (1 + 119909)V 119875(120583V)119899 (119909)

sdot 119875(120588120590)119898 (119909) 1198891199091198891205851 119889120585119903

= Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899)119898119899sdot infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) 22119896 (119896)2times 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) 119911120585119894119894 times int1

minus1(1

minus 119909)1+120575+sum119903119894=1 119897119894120585119894+2119896minus1 (1 + 119909)1+Vminus1 1198891199091198891205851 119889120585119903

(30)

Now using (21) we have the following

1198687= Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899)119898119899sdot infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) 22119896 (119896)2times 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903)sdot 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 2120575+sum119903119894=1 119897119894120585119894+2119896+V+1 times 119861(120575 + 119903sum119894=1

119897119894120585119894 + 2119896

+ 1 V + 1)1198891205851 119889120585119903 = 2120575+V+1

sdot Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899) Γ (V + 1)119898119899

times infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) (119896)2

times 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903)sdot 119903prod119894=1

120601119894 (120585119894) (2119897119894119911119894)120585119894 Γ (120575 + sum119903119894=1 119897119894120585119894 + 2119896 + 1)

Γ (120575 + sum119903119894=1 119897119894120585119894 + 2119896 + V + 2)1198891205851

119889120585119903

(31)

Finally we arrive at

1198687 = 2120575+V+1 Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899) Γ (V + 1)119898119899 times infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) (119896)2

times alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

2119897111991112119897119903119911119903

| (minus120575 minus 2119896 1198971 119897119903) 119860 119861(minus1 minus 120575 minus 2119896 minus V 1198971 119897119903) 119862 119863

]]]]]

(32)

Thus 120575 gt 0R(V) gt minus1 and |arg 119911| lt (12)120587Ω

1198688 = int1

minus1(1 minus 119909)120588 (1 + 119909)120590 119875(120583V)

119899 (119909)

times alefsym0119899119880119901119894119902119894 120591119894119877119881119882

(1199111 (1 minus 119909)1198971 (1 + 119909)ℎ1 119911119903 (1

minus 119909)119897119903 (1 + 119909)ℎ119903) 119889119909 = Γ (1 + 120583 + 119899)119899

sdot infinsum119896=0

(minus119899)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120583 + 119896) 2119896119896 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851

120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 times int1

minus1(1 minus 119909)1+120588+sum119903119894=1 119897119894120585119894+119896minus1

sdot (1 + 119909)1+120590+sum119903119894=1 ℎ119894120585119894minus1 1198891199091198891205851 119889120585119903(33)

Using (21) we have the following

6 Journal of Applied Mathematics

1198688 = 2120588+120590+1 Γ (1 + 120583 + 119899)119899

infinsum119896=0

(minus119899)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120583 + 119896) 119896times 1(2120587120596)119903 int1198711 sdot sdot sdotsdot int

119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (2119897119894+ℎ119894119911119894)120585119894

times Γ (1 + 120588 + 119896 + sum119903119894=1 119897119894120585119894) Γ (1 + 120590 + sum119903

119894=1 ℎ119894120585119894)Γ (2 + 120588 + 120590 + 119896 + sum119903119894=1 (119897119894 + ℎ119894) 120585119894) 1198891205851

119889120585119903(34)

Finally rewriting the above equation by virtue of (1) we arriveat

1198688 = 2120588+120590+1 Γ (1 + 120583 + 119899)119899

infinsum119896=0

(minus119899)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120583 + 119896) 119896

times alefsym0119899+2119880119901119894+2119902119894+1120591119894 119877119881119882

[[[[[

21198971+ℎ111991112119897119903+ℎ119903119911119903

| (minus120588 minus 119896 1198971 119897119903) (minus120590 ℎ1 ℎ119903) 119860 119861(minus1 minus 120588 minus 120590 minus 119896) (1198971 + ℎ1) (119897119903 + ℎ119903)) 119862 119863

]]]]]

(35)

ThusR(V) gt minus1R(120583) gt minus1 and |arg 119911| lt (12)120587Ω4 Integrals Involving Multivariable Aleph-Function with Legendre Function

The solution of Legendre differential equation in the form ofGauss hypergeometric type is as follows

119901120583V (119909) = 1Γ (1 minus 120583) (119909 + 1119909 minus 1)

12120583

sdot 21198651 [minusV V + 1 1 minus 120583 1 minus 1199092 ] |1 minus 119909| lt 2

(36)

Here119901120583V (119909) is known as the Legendre function of the first kind

[16]

Next we derive the integrals with Legendre function

1198689 = int1

0119909120590minus1 (1 minus 1199092)1205832 119901120583

V (119909)sdot alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205881 119911119903119909120588119903] 119889119909 = 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

0119909120590+sum119903119894=1 120588119894120585119894minus1 (1 minus 1199092)1205832 119875120583

V (119909) 1198891199091198891205851 119889120585119903

(37)

Next we use the following formula ([16] Sec 312) for 120583 isinNR(120590) gt 0

int1

0119909120590minus1 (1 minus 1199092)1205832 119901120583

V (119909) 119889119909 = (minus1)120583 2minus120590minus12058312058712Γ (120590) Γ (1 + 120583 + V)Γ (1 minus 120583 + V) Γ (12 + 1205902 + 1205832 minus V2) Γ (1 + 1205902 + 1205832 + V2) (38)

Now applying (38) we obtain

1198689 = 2minus120590minus120583 (minus1)120583 12058712 Γ (1 + 120583 + V)Γ (1 minus 120583 + V) 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) (2minus120588119894119911119894)120585119894

Journal of Applied Mathematics 7

times Γ (120590 + sum119903119894=1 120588119894120585119894)Γ (12 + 1205902 + 1205832 minus V2 + (sum119903

119894=1 120588119894120585119894) 2) Γ (1 + 1205902 + 1205832 + V2 + (sum119903119894=1 120588119894120585119894) 2)1198891205851 119889120585119903

(39)

1198689 = 2minus120590minus120583 (minus1)120583 12058712 Γ (1 + 120583 + V)Γ (1 minus 120583 + V)

sdot alefsym0119899+1119880119901119894+1119902119894+2120591119894 119877119881119882

[[[[[

2minus120588111991112minus120588119903119911119903

| (1 minus 120590 1205881 120588119903) 119860 119861(12 minus 1205832 + V2 minus 1205902 12058812 1205881199032 ) (minus1205832 minus V2 minus 1205902 12058812 1205881199032 ) 119862 119863

]]]]]

(40)

Thus |arg (119911)| lt (12)120587Ω 120590 gt 0 and 120583 isin N cup 011986810 = int1

0119909120590minus1 (1 minus 1199092)minus1205832 119901120583

V (119909)sdot alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205881 119911119903119909120588119903] 119889119909 = 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

0119909120590+sum119903119894=1 120588119894120585119894minus1 (1 minus 1199092)minus1205832 119875120583

V (119909) 1198891199091198891205851

119889120585119903(41)

Next we use the following formula ([16] Sec 312) for 120583 isinNR(120590) gt 0

int1

0119909120590minus1 (1 minus 1199092)minus1205832 119901120583

V (119909) 119889119909= 2minus120590+12058312058712Γ (120590)Γ (12 + 1205902 minus 1205832 minus V2) Γ (1 + 1205902 minus 1205832 minus V2)

(42)

Using (42) we obtain

11986810 = 2minus120590+12058312058712 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (2minus120588119894119911119894)120585119894

times Γ (120590 + sum119903119894=1 120588119894120585119894)Γ (12 + 1205902 minus 1205832 minus V2 + (sum119903

119894=1 120588119894120585119894) 2) Γ (1 + 1205902 minus 1205832 minus V2 + (sum119903119894=1 120588119894120585119894) 2)1198891205851 119889120585119903

(43)

11986810 = 2minus120590+12058312058712alefsym0119899+1119880119901119894+1119902119894+2120591119894119877119881119882

[[[[[

2minus120588111991112minus120588119903119911119903

| (1 minus 120590 1205881 120588119903) 119860 119861(12 + 1205832 + V2 minus 1205902 12058812 1205881199032 ) (1205832 + V2 minus 1205902 12058812 1205881199032 ) 119862 119863

]]]]] (44)

Thus |arg (119911)| lt (12)120587ΩR(120590) gt 0 andR(120583) isin N cup 0

5 Integrals Involving Multivariable Aleph-Function and Bessel-Maitland Function

The Bessel-Maitland function is defined by Kiryakova [17] asfollows

119869120583V (119909) = 120601 (120583 V + 1 119909) = infinsum119896=0

1Γ (120583119896 + V + 1) (minus119909)119896

119896 120583 gt minus1 119909 isin C

(45)

11986811 = intinfin

0119909120588119869120583V (119909)

sdot alefsym0119899119880119901119894119902119894120591119894 119877119881119882

[11991111199091205901 119911119903119909120590119903] 119889119909 119889119909= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585i) 119911120585119894119894

sdot intinfin

0119909120588+sum119903119894=1 120590119894120585119894119869120583V (119909) 119889119909 1198891205851 119889120585119903

(46)

Now using the following formula from [18]

8 Journal of Applied Mathematics

intinfin

0119909120588119869120583V (119909) 119889119909 = Γ (120588 + 1)

Γ (1 + V minus 120583 minus 120583120588)(R (120588) gt minus1 0 lt 120583 lt 1)

(47)and further applying (47) we obtain

11986814 = 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 Γ (1 + 120588 + sum119903119894=1 120590119894120585119894)Γ (1 + V minus 120583 minus 120583120588 minus 120583sum119903

119894=1 120590119894120585119894)1198891205851 119889120585119903 (48)

11986814 = alefsym0119899+1119880119901119894+2119902119894 120591119894119877119881119882

[[[[[

1199111119911119903

| (minus120588 1205901 120590119903) (1 + V minus 120583 minus 120583120588 1205831205901 120583120590119903) 119860 119861119862 119863

]]]]] (49)

Thus |arg 119911| lt (12)120587Ω 120590 minus 120583120590 gt 0 120590 gt 0 0 lt 120583 lt 1 andR(120588 + 1) gt 06 Integrals Involving Multivariable Aleph-Function and General Class of Polynomials

The general class of polynomials 1198781198981 1198981199031198991 119899119903(119909) defined by

Srivastava [19] is as follows

1198781198981 1198981199031198991 119899119903(119909) = [11989911198981]sum

119897119894=0

sdot sdot sdot [119899119903119898119903]sum119897119903=0

119903prod119894=1

(minus119899119894)119898119894119897119894119897119894 119860119899119894 119897119894119909119897119894 (50)

Here 1198991 1198992 119899119903 isin N0 1198981 1198982 119898119903 isin N and thecoefficients 119860119899119894 119897119894

(119899119894 119897119894 ge 0) are any constant numbers Bysuitable restriction of the coefficient 119860119899119894 119897119894

the general classof polynomials has various special cases These include theJacobi polynomials the Laguerre polynomials the Besselpolynomials the Hermite polynomials the Gould-Hopperpolynomials and the Brafman polynomials ([20] p 158-161)

Next we establish the following integral

11986812 = int1

minus1(1 minus 119909)120588minus1 (1 + 119909)120590minus1 1198781198981 11989811990310158401198991 1198991199031015840

[119910 (1 minus 119909)120583 (1+ 119909)V] times alefsym0119899119880

119901119894119902119894 120591119894119877119881119882(1199111 (1 minus 119909)ℎ1

sdot (1 + 119909)1198961 119911119903 (1 minus 119909)ℎ119903 (1 + 119909)119896119903) 119889119909= [11989911198981]sum

1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895 119897119895119910119897119895 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 sdot sdot sdot 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

minus1(1 minus 119909)120588+sum119903119894=1 ℎ119894120585119894+120583sum1199031015840119895=1 119897119895minus1

sdot (1 + 119909)120590+sum119903119894=1 119896119894120585119894+Vsum1199031015840119895=1 119897119895minus1 1198891199091198891205851 119889120585119903

(51)

By applying (21) we have

11986812 = 2120588+120590minus1[11989911198981]sum1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895119897119895(2120583+V119910)119897119895 times 1(2120587120596)119903 int1198711 sdot sdot sdot

sdot int119871119903

120595 (1205851 sdot sdot sdot 120585119903) 119903prod119894=1

120601119894 (120585119894) (2ℎ119894+119896119894119911119894)120585119894 times Γ (120588 + sum119903119894=1 ℎ119894120585119894 + 120583sum1199031015840

119895=1 119897119895) Γ (120590 + sum119903119894=1 119896119894120585119894 + Vsum1199031015840

119895=1 119897119895)Γ (120588 + 120590 + sum119903

119894=1 (ℎ119894 + 119896119894) 120585119894 + (120583 + V)sum1199031015840

119895=1 119897119895) 1198891205851 119889120585119903(52)

11986812 = 2120588+120590minus1[11989911198981]sum1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895119897119895(2120583+V119910)119897119895

sdot alefsym0119899+1119880119901119894+2119902119894 120591119894119877119881119882

[[[[[[

2ℎ1+119896111991112ℎ119903+119896119903119911119903

| (1 minus 120588 minus 120583 1199031015840sum119895=1

119897119895 ℎ1 ℎ119903) (1 minus 120590 minus V1199031015840sum119895=1

119897119895 1198961 119896119903) 119860 119861(1 minus 120588 minus 120590 minus (120583 + V) 1199031015840sum

119895=1119897119895 (ℎ1 + 1198961) (ℎ119903 + 119896119903)) 119862 119863

]]]]]]

(53)

Journal of Applied Mathematics 9

These converge under the following conditions

(1) |arg 119911| lt (12)120587Ω(2) 120588 ge 1 120590 ge 1 120583 ge 0 V ge 0 ℎ ge 0 119896 ge 0 (ℎ and 119896 are

not both zero simultaneously)(3) R(120588) + ℎmin[R(119887119895119861119895)] gt 0 and R(120590) +119896min[R(119887119895119861119895)] gt 0

7 Special Cases

(i)By setting 120575 by 120578minus1 and120588 = 120590 = 120583 = V = 0 (32) transformsto the following integral

11986813 = 2120578 infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 119898)119896 (1 + 119899)119896Γ (1 + 119896) Γ (1 + 119896) (119896)2

times alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

211989711991112119897119911119903

| (1 minus 120578 minus 2119896 1198971 119897119903) 119860 119861(minus120578 minus 2119896 1198971 119897119903) 119862 119863

]]]]](54)

Thus |arg 119911| lt (12)120587Ω(ii)When we substitute 120588 by 120588 minus 1 and 120590 by 120590 minus 1 and put120583 = V = 0 integral (35) transforms to the following

11986814 = 2120588+120590minus1 infinsum119896=0

(minus119899)119896 (1 + 119899)119896Γ (1 + 119896) 119896 times alefsym0119899+2119880119901119894+2119902119894+1120591119894 119877119881119882

[[[[[

21198971+ℎ111991112119897119903+ℎ119903119911119903

| (1 minus 120588 minus 119896 1198971 119897119903) (1 minus 120590 ℎ1 ℎ119903) 119860 119861(1 minus 120588 minus 120590 minus 119896 (1198971 + ℎ1) (119897119903 + ℎ119903)) 119862 119863

]]]]] (55)

Thus |arg 119911| lt (12)120587Ω8 Concluding Remarks

In the present paper we evaluated new integrals involving themultivariable Aleph-function with certain special functionsCertain special cases of integrals that follow Remarks 1ndash3have been investigated in the literature by a number ofauthors [21ndash24]) with different arguments Therefore theresults presented in this paper are easily converted in termsof a similar type of new interesting integrals with differentarguments after some suitable parameter substitutions Inthis sequel one can obtain integral representation of moregeneralized special function which has many applications inphysics and engineering science

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C K Sharma and S S Ahmad ldquoOn the multivariable I-functionrdquo Acta Ciencia Indica Mathematics vol 20 no 2 pp113ndash116 1994

[2] Y N Prasad ldquoMultivariable I-functionrdquo Vijnana ParishadAnusandhan Patrika vol 29 pp 231ndash235 1986

[3] C K Sharma and G K Mishra ldquoExponential Fourier series forthe multivariable I-functionrdquo Acta Ciencia Indica Mathematicsvol 21 no 4 pp 495ndash501 1995

[4] H M Srivastava and R Panda ldquoSome expansion theoremsand generating relations for the H function of several complex

variablesrdquo Commentarii Mathematici Universitatis Sancti Paulivol 24 no 2 pp 119ndash137 1975

[5] N Sudland B Baumann andT FNonnenmacher ldquoOpen prob-lem Who knows about the Aleph () -functions rdquo FractionalCalculus and Applied Analysis vol 1 no 4 pp 401-402 1998

[6] D Kumar S D Purohit and J Choi ldquoGeneralized fractionalintegrals involving product of multivariable H-function and ageneral class of polynomialsrdquo Journal of Nonlinear Sciences andApplications vol 09 no 01 pp 8ndash21 2016

[7] R K Saxena and T K Pogany ldquoOn fractional integrationformulae for Aleph functionsrdquo Applied Mathematics and Com-putation vol 218 no 3 pp 985ndash990 2011

[8] R K Saxena J Ram and D Kumar ldquoGeneralized fractionalintegral of the product of two Aleph-functionsrdquo Applicationsand Applied Mathematics An International Journal (AAM) vol8 no 2 pp 631ndash646 2013

[9] R K Saxena J Ram and D L Suthar ldquoUnified fractionalderivative formulas for the multivariable H-functionrdquo VijnanaParishad Anusandhan Patrika vol 49 no 2 pp 159ndash175 2006

[10] D L Suthar and P Agarwal ldquoGeneralized mittag -leffler func-tion and the multivariable h-function involving the generalizedmellin-barnes contour integralsrdquoCommunications inNumericalAnalysis vol 2017 no 1 pp 25ndash33 2017

[11] D L Suthar B Debalkie and M Andualem ldquoModifiedSaigo fractional integral operators involving multivariableH-function and general class of multivariable polynomialsrdquoAdvances in Difference Equations vol 2019 no 1 article no 2132019

[12] D L Suthar H Habenom and H Tadesse ldquoCertain integralsinvolving aleph function and Wrightrsquos generalized hypergeo-metric functionrdquo Acta Universitatis Apulensis no 52 pp 1ndash102017

[13] D L Suthar G Reddy and B Shimelis ldquoOn Certain UnifiedFractional Integrals Pertaining to Product of Srivastavarsquos Poly-nomials and N-Functionrdquo International Journal of MathematicsTrends and Technology vol 47 no 1 pp 66ndash73 2017

[14] E D Rainville Special Functions The Macmillan CompanyNew York NY USA 1960

10 Journal of Applied Mathematics

[15] G Szego Orthogonal Polynomials American MathematicalSociety Colloquium Publications vol 23 American Mathemati-cal Society Providence RI USA 1959

[16] A Erdersquolyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol I McGraw-Hill NewYork-Toronto-London 1953

[17] V S Kiryakova Generalized Fractional Calculus and Applica-tions vol 301 of Pitman Research Notes in Mathematics UnitedStates with JohnWiley amp Sons Longman Scientific ampTechnicalNew York NY USA 1994

[18] V P Saxena ldquoFormal solution of certain new pair of dualintegral equations involving H-functionsrdquo Proceedings of theNational Academy of Sciences vol A52 no 3 pp 366ndash375 1982

[19] H M Srivastava ldquoA multilinear generating function for theKonhauser sets of biorthogonal polynomials suggested by theLaguerre polynomialsrdquo Pacific Journal of Mathematics vol 117no 1 pp 183ndash191 1985

[20] H M Srivastava and N P Singh ldquoThe integration of certainproducts of the multivariableH-function with a general class ofpolynomialsrdquo Rendiconti del Circolo Matematico di Palermo vol 32 no 2 pp 157ndash187 1983

[21] F Ayant ldquoThe integration of certains products of special func-tions and multivariable Aleph-functionsrdquo International Journalof Mathematics Trends and Technology vol 31 no 2 pp 101ndash1122016

[22] D Kumar R K Saxena and J Ram ldquoFinite integral formulasinvolving aleph functionrdquo Boletim da Sociedade Paranaense deMatematica vol 36 no 1 pp 177ndash193 2018

[23] D L Suthar S Agarwal and D Kumar ldquoCertain integralsinvolving the product of Gaussian hypergeometric function andAleph functionrdquoHonamMathematical Journal vol 41 no 1 pp1ndash17 2019

[24] D L Suthar and B Debalkie ldquoA new class of integral relationinvolving Aleph-functionsrdquo Surveys in Mathematics and itsApplications vol 12 pp 193ndash201 2017

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Journal of Applied Mathematics 3

1198682 = int1

0119909120588minus1 (1 minus 119909)120590minus1 alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205961 119911119903119909120596119903] 119889119909

= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894sdot int1

0119909120588+sum119903119894=1 120596119894120585119894minus1 (1 minus 119909)120590minus1 1198891199091198891205851 119889120585119903 = 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 119861(120588 + 119903sum119894=1

120596119894120585119894 120590)1198891205851 sdot sdot sdot 119889120585119903= Γ (120590) 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) 119911120585119894119894sdot Γ (120588 + sum119903

119894=1 120585119894)Γ (120588 + 120590 + sum119903119894=1 120585119894)1198891205851 sdot sdot sdot 119889120585119903 = Γ (120590)

sdot alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

1199111119911119903

| (1 minus 120588 1205961 120596119903) 119860 119861(1 minus 120588 minus 120590 1205961 120596119903) 119862 119863

]]]]]

(12)

1198683 = intinfin

1119909minus120588 (119909 minus 1)120590minus1 alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205961 119911119903119909120596119903] 119889119909

= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894sdot intinfin

1119909minus120588+sum119903119894=1 120596119894120585119894 (119909 minus 1)120590minus1 1198891199091198891205851 119889120585119903

(13)

Putting 119909 = 119905 + 1 we have 119889119909 = 119889119905 and we use the followingrelation

Γ (120572) Γ (120573)Γ (120572 + 120573) = intinfin

0119909120572minus1 (119909 + 1)minus(120572+120573) 119889119909 = intinfin

0119909120573minus1 (119909

+ 1)minus(120572+120573) 119889119909(14)

1198683 = 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894sdot intinfin

0119905120590minus1 (1 + 119905)minus(120588minussum119903119894=1 120596119894120585119894) 119889119905 1198891205851 119889120585119903 = Γ (120590)

sdot 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903)sdot 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 Γ (120588 minus 120590 minus sum119903119894=1 120596119894120585119894)Γ (120588 minus sum119903

119894=1 120596119894120585119894) 1198891205851 sdot sdot sdot 119889120585119903 = Γ (120590)

sdot alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

1199111119911119903

| (120588 1205961 120596119903) 119860 119861(120588 minus 120590 1205961 120596119903) 119862 119863

]]]]]

(15)

1198684 = intinfin

0119909120588minus1 (119909 + 120573)minus120590 alefsym0119899119880

119901119894119902119894120591119894119877119881119882[11991111199091205961 119911119903119909120596119903] 119889119909

= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894sdot intinfin

0119909120588+sum119903119894=1 120596119894120585119894minus1 (119909 + 120573)minus120590 1198891199091198891205851 119889120585119903

(16)

Setting 119909 = 120573119905 implies 119889119909 = 120573119889119905 and then we obtain thefollowing

= 120573120588minus120590

(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (120573120596119894119911119894)120585119894

sdot intinfin

0119905120588+sum119903119894=1 120596119894120585119894minus1 (1 + 119905)minus120590 119889119905 1198891205851 119889120585119903

(17)

By applying (14) we have the following

= 120573120588minus120590

Γ (120590) 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (120573120596119894119911119894)120585119894

times Γ(120588 + 119903sum119894=1

120596119894120585119894)Γ(120590 minus 120588 minus 119903sum119894=1

120596119894120585119894)1198891205851 119889120585119903(18)

= 120573120588minus120590

Γ (120590)

sdot alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

12057312059611199111120573120596119903119911119903

| (1 minus 120588 1205961 120596119903) 119860 119861(120590 minus 120588 1205961 120596119903) 119862 119863

]]]]](19)

1198685 = int1

minus1(1 minus 119909)120588 (1 + 119909)120590

sdot alefsym0119899119880119901119894119902119894 120591119894119877119881119882

[1199111 (1 minus 119909)1205831 119911119903 (1 minus 119909)120583119903] 119889119909= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) 119911120585119894119894sdot int1

minus1(1 minus 119909)120588+sum119903119894=1 120583119894120585119894 (1 + 119909)120590 1198891199091198891205851 119889120585119903

(20)

Now we use the following formula ([14] p261)

int1

minus1(1 minus 119909)119899+120572 (1 + 119909)119899+120573 119889119909= 22119899+120572+120573+1119861 (1 + 120572 + 119899 1 + 120573 + 119899)

(21)

Hence we arrive at

= 2120588+120590+1Γ (1 + 120590) 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (2120583119894119911119894)120585119894 times Γ (1 + 120588 + sum119903119894=1 120583119894120585119894)Γ (2 + 120588 + 120590 + 120583sum119903

119894=1 120583119894120585119894)1198891205851 119889120585119903 (22)

= 2120588+120590+1Γ (1 + 120590)alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

2120583111991112120583119903119911119903

| (minus120588 1205831 120583119903) 119860 119861(minus1 minus 120590 minus 120588 1205831 120583119903) 119862 119863

]]]]] (23)

4 Journal of Applied Mathematics

3 Integrals Involving Multivariable Aleph-Function with Jacobi Polynomials

The Jacobi polynomial ([15] 4212)) with parameters 120572 120573 isinR is defined by

119901(120572120573)119899 (119909)= (1 + 120572)119899119899 21198651 [minus119899 1 + 120572 + 120573 + 119899 1 + 120572 1 minus 1199092 ]

119899 ge 1(24)

where 21198651[] is the classical hypergeometric function Bysubstituting 120572 = 120573 = 0 the Jacobi polynomial (24) reducesto Lagrange polynomial ( [14] p 157) as

119901(120572120573)119899 (1) = (1 + 120572)119899119899 (25)

In this section we derive integral formulas involving multi-variable Aleph-functions multiplied by Jacobi polynomials

1198686= int1

minus1119909120582 (1

minus 119909)120572 (1 + 119909)120583 119875(120572120573)119899 (119909)alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[1199111 (1 + 119909)1198971

119911119903 (1 + 119909)119897119903] 119889119909

= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894)

sdot 119911120585119894119894 times int1

minus1119909120582 (1 minus 119909)120572 (1 + 119909)120583+sum119903119894=1 119897119894120585119894

sdot 119875(120572120573)119899 (119909) 119889119909 1198891205851 119889120585119903

(26)

Next we use the following formula

int1

minus1119909120582 (1 minus 119909)120572 (1 + 119909)120583 119901(120572120573)

119899 (119909) 119889119909= (minus1)119899 2120572+120583+1Γ (120583 + 1) Γ (119899 + 120572 + 1) Γ (120583 + 120573 + 1)

119899Γ (120583 + 120573 + 119899 + 1) Γ (120583 + 120572 + 119899 + 2)times 31198652 [ minus120582 120583 + 120573 + 1 120583 + 1

120583 + 120573 + 119899 + 1 120583 + 120572 + 119899 + 2 1] (27)

where 120572 gt minus1 and 120573 gt minus1 Also 31198652 is the special case ofgeneralized hypergeometric series

Then we have the following

= 1(2120587120596)119903sdot int

1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) rprod119894=1

120601119894 (120585119894) 119911120585119894119894 (minus1)119899 2120572+120583+sum119903119894=1 119897119894120585119894+1119899times Γ (120583 + sum119903

119894=1 119897119894120585119894 + 1) Γ (119899 + 120572 + 1) Γ (120583 + sum119903119894=1 119897119894120585119894 + 120573 + 1)

Γ (120583 + sum119903119894=1 119897119894120585119894 + 120573 + 119899 + 1) Γ (120583 + sum119903

119894=1 119897119894120585119894 + 120572 + 119899 + 2)

times 31198652

[[[[

minus120582 120583 + 119903sum119894=1

119897119894120585119894 + 120573 + 1 120583 + 119903sum119894=1

119897119894120585119894 + 1120583 + 119903sum

119894=1119897119894120585119894 + 120573 + 119899 + 1 120583 + 119903sum

119894=1119897119894120585119894 + 120572 + 119899 + 2 1]]]

]1198891205851

119889120585119903

(28)

Using the definition of hypergeometric function and somesimplifications the above expression becomes

1198686 = (minus1)119899 2120572+120583+1Γ (120572 + 119899 + 1)119899infinsum119896=0

(minus120582)119896 1119896119896 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) times (2119897119894119911119894)120585119894 Γ (120583 + sum119903119894=1 119897119894120585119894 + 120573 + 119896 + 1) Γ (120583 + sum119903

119894=1 119897119894120585119894 + 119896 + 1)Γ (120583 + sum119903

119894=1 119897119894120585119894 + 120573 + 119899 + 119896 + 1) Γ (120583 + sum119903119894=1 119897119894120585119894 + 120572 + 119899 + 119896 + 2)1198891205851

119889120585119903 = (minus1)119899 2120572+120583+1Γ (120572 + 119899 + 1)119899infinsum119896=0

(minus120582)119896 1119896119896

sdot alefsym0119899+2119880119901119894+2119902119894+2120591119894 119877119881119882

[[[[[

2119897111991112119897119903119911119903

| (minus120583 minus 120573 minus 119896 1198971 119897119903) (minus120583 minus 119896 1198971 119897119903) 119860 119861(minus120583 minus 120573 minus 119899 minus 119896 1198971 119897119903) (minus1 minus 120583 minus 120572 minus 119899 minus 119896 1198971 119897119903) 119862 119863

]]]]]

(29)

Journal of Applied Mathematics 5

Thus 120572 gt minus1 120573 gt minus1R(120582) gt minus1 and |arg 119911| lt (12)120587Ω

1198687= int1

minus1(1

minus 119909)120575 (1 + 119909)V 119875(120583V)119899 (119909) 119875(120588120590)

119898 (119909) times alefsym0119899119880119901119894119902119894 120591119894119877119881119882

[1199111 (1minus 119909)1198971 119911119903 (1 minus 119909)119897119903] 119889119909= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) 119911120585119894119894 times int1

minus1(1

minus 119909)120575+sum119903119894=1 119897119894120585119894 (1 + 119909)V 119875(120583V)119899 (119909)

sdot 119875(120588120590)119898 (119909) 1198891199091198891205851 119889120585119903

= Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899)119898119899sdot infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) 22119896 (119896)2times 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) 119911120585119894119894 times int1

minus1(1

minus 119909)1+120575+sum119903119894=1 119897119894120585119894+2119896minus1 (1 + 119909)1+Vminus1 1198891199091198891205851 119889120585119903

(30)

Now using (21) we have the following

1198687= Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899)119898119899sdot infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) 22119896 (119896)2times 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903)sdot 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 2120575+sum119903119894=1 119897119894120585119894+2119896+V+1 times 119861(120575 + 119903sum119894=1

119897119894120585119894 + 2119896

+ 1 V + 1)1198891205851 119889120585119903 = 2120575+V+1

sdot Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899) Γ (V + 1)119898119899

times infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) (119896)2

times 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903)sdot 119903prod119894=1

120601119894 (120585119894) (2119897119894119911119894)120585119894 Γ (120575 + sum119903119894=1 119897119894120585119894 + 2119896 + 1)

Γ (120575 + sum119903119894=1 119897119894120585119894 + 2119896 + V + 2)1198891205851

119889120585119903

(31)

Finally we arrive at

1198687 = 2120575+V+1 Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899) Γ (V + 1)119898119899 times infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) (119896)2

times alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

2119897111991112119897119903119911119903

| (minus120575 minus 2119896 1198971 119897119903) 119860 119861(minus1 minus 120575 minus 2119896 minus V 1198971 119897119903) 119862 119863

]]]]]

(32)

Thus 120575 gt 0R(V) gt minus1 and |arg 119911| lt (12)120587Ω

1198688 = int1

minus1(1 minus 119909)120588 (1 + 119909)120590 119875(120583V)

119899 (119909)

times alefsym0119899119880119901119894119902119894 120591119894119877119881119882

(1199111 (1 minus 119909)1198971 (1 + 119909)ℎ1 119911119903 (1

minus 119909)119897119903 (1 + 119909)ℎ119903) 119889119909 = Γ (1 + 120583 + 119899)119899

sdot infinsum119896=0

(minus119899)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120583 + 119896) 2119896119896 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851

120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 times int1

minus1(1 minus 119909)1+120588+sum119903119894=1 119897119894120585119894+119896minus1

sdot (1 + 119909)1+120590+sum119903119894=1 ℎ119894120585119894minus1 1198891199091198891205851 119889120585119903(33)

Using (21) we have the following

6 Journal of Applied Mathematics

1198688 = 2120588+120590+1 Γ (1 + 120583 + 119899)119899

infinsum119896=0

(minus119899)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120583 + 119896) 119896times 1(2120587120596)119903 int1198711 sdot sdot sdotsdot int

119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (2119897119894+ℎ119894119911119894)120585119894

times Γ (1 + 120588 + 119896 + sum119903119894=1 119897119894120585119894) Γ (1 + 120590 + sum119903

119894=1 ℎ119894120585119894)Γ (2 + 120588 + 120590 + 119896 + sum119903119894=1 (119897119894 + ℎ119894) 120585119894) 1198891205851

119889120585119903(34)

Finally rewriting the above equation by virtue of (1) we arriveat

1198688 = 2120588+120590+1 Γ (1 + 120583 + 119899)119899

infinsum119896=0

(minus119899)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120583 + 119896) 119896

times alefsym0119899+2119880119901119894+2119902119894+1120591119894 119877119881119882

[[[[[

21198971+ℎ111991112119897119903+ℎ119903119911119903

| (minus120588 minus 119896 1198971 119897119903) (minus120590 ℎ1 ℎ119903) 119860 119861(minus1 minus 120588 minus 120590 minus 119896) (1198971 + ℎ1) (119897119903 + ℎ119903)) 119862 119863

]]]]]

(35)

ThusR(V) gt minus1R(120583) gt minus1 and |arg 119911| lt (12)120587Ω4 Integrals Involving Multivariable Aleph-Function with Legendre Function

The solution of Legendre differential equation in the form ofGauss hypergeometric type is as follows

119901120583V (119909) = 1Γ (1 minus 120583) (119909 + 1119909 minus 1)

12120583

sdot 21198651 [minusV V + 1 1 minus 120583 1 minus 1199092 ] |1 minus 119909| lt 2

(36)

Here119901120583V (119909) is known as the Legendre function of the first kind

[16]

Next we derive the integrals with Legendre function

1198689 = int1

0119909120590minus1 (1 minus 1199092)1205832 119901120583

V (119909)sdot alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205881 119911119903119909120588119903] 119889119909 = 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

0119909120590+sum119903119894=1 120588119894120585119894minus1 (1 minus 1199092)1205832 119875120583

V (119909) 1198891199091198891205851 119889120585119903

(37)

Next we use the following formula ([16] Sec 312) for 120583 isinNR(120590) gt 0

int1

0119909120590minus1 (1 minus 1199092)1205832 119901120583

V (119909) 119889119909 = (minus1)120583 2minus120590minus12058312058712Γ (120590) Γ (1 + 120583 + V)Γ (1 minus 120583 + V) Γ (12 + 1205902 + 1205832 minus V2) Γ (1 + 1205902 + 1205832 + V2) (38)

Now applying (38) we obtain

1198689 = 2minus120590minus120583 (minus1)120583 12058712 Γ (1 + 120583 + V)Γ (1 minus 120583 + V) 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) (2minus120588119894119911119894)120585119894

Journal of Applied Mathematics 7

times Γ (120590 + sum119903119894=1 120588119894120585119894)Γ (12 + 1205902 + 1205832 minus V2 + (sum119903

119894=1 120588119894120585119894) 2) Γ (1 + 1205902 + 1205832 + V2 + (sum119903119894=1 120588119894120585119894) 2)1198891205851 119889120585119903

(39)

1198689 = 2minus120590minus120583 (minus1)120583 12058712 Γ (1 + 120583 + V)Γ (1 minus 120583 + V)

sdot alefsym0119899+1119880119901119894+1119902119894+2120591119894 119877119881119882

[[[[[

2minus120588111991112minus120588119903119911119903

| (1 minus 120590 1205881 120588119903) 119860 119861(12 minus 1205832 + V2 minus 1205902 12058812 1205881199032 ) (minus1205832 minus V2 minus 1205902 12058812 1205881199032 ) 119862 119863

]]]]]

(40)

Thus |arg (119911)| lt (12)120587Ω 120590 gt 0 and 120583 isin N cup 011986810 = int1

0119909120590minus1 (1 minus 1199092)minus1205832 119901120583

V (119909)sdot alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205881 119911119903119909120588119903] 119889119909 = 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

0119909120590+sum119903119894=1 120588119894120585119894minus1 (1 minus 1199092)minus1205832 119875120583

V (119909) 1198891199091198891205851

119889120585119903(41)

Next we use the following formula ([16] Sec 312) for 120583 isinNR(120590) gt 0

int1

0119909120590minus1 (1 minus 1199092)minus1205832 119901120583

V (119909) 119889119909= 2minus120590+12058312058712Γ (120590)Γ (12 + 1205902 minus 1205832 minus V2) Γ (1 + 1205902 minus 1205832 minus V2)

(42)

Using (42) we obtain

11986810 = 2minus120590+12058312058712 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (2minus120588119894119911119894)120585119894

times Γ (120590 + sum119903119894=1 120588119894120585119894)Γ (12 + 1205902 minus 1205832 minus V2 + (sum119903

119894=1 120588119894120585119894) 2) Γ (1 + 1205902 minus 1205832 minus V2 + (sum119903119894=1 120588119894120585119894) 2)1198891205851 119889120585119903

(43)

11986810 = 2minus120590+12058312058712alefsym0119899+1119880119901119894+1119902119894+2120591119894119877119881119882

[[[[[

2minus120588111991112minus120588119903119911119903

| (1 minus 120590 1205881 120588119903) 119860 119861(12 + 1205832 + V2 minus 1205902 12058812 1205881199032 ) (1205832 + V2 minus 1205902 12058812 1205881199032 ) 119862 119863

]]]]] (44)

Thus |arg (119911)| lt (12)120587ΩR(120590) gt 0 andR(120583) isin N cup 0

5 Integrals Involving Multivariable Aleph-Function and Bessel-Maitland Function

The Bessel-Maitland function is defined by Kiryakova [17] asfollows

119869120583V (119909) = 120601 (120583 V + 1 119909) = infinsum119896=0

1Γ (120583119896 + V + 1) (minus119909)119896

119896 120583 gt minus1 119909 isin C

(45)

11986811 = intinfin

0119909120588119869120583V (119909)

sdot alefsym0119899119880119901119894119902119894120591119894 119877119881119882

[11991111199091205901 119911119903119909120590119903] 119889119909 119889119909= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585i) 119911120585119894119894

sdot intinfin

0119909120588+sum119903119894=1 120590119894120585119894119869120583V (119909) 119889119909 1198891205851 119889120585119903

(46)

Now using the following formula from [18]

8 Journal of Applied Mathematics

intinfin

0119909120588119869120583V (119909) 119889119909 = Γ (120588 + 1)

Γ (1 + V minus 120583 minus 120583120588)(R (120588) gt minus1 0 lt 120583 lt 1)

(47)and further applying (47) we obtain

11986814 = 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 Γ (1 + 120588 + sum119903119894=1 120590119894120585119894)Γ (1 + V minus 120583 minus 120583120588 minus 120583sum119903

119894=1 120590119894120585119894)1198891205851 119889120585119903 (48)

11986814 = alefsym0119899+1119880119901119894+2119902119894 120591119894119877119881119882

[[[[[

1199111119911119903

| (minus120588 1205901 120590119903) (1 + V minus 120583 minus 120583120588 1205831205901 120583120590119903) 119860 119861119862 119863

]]]]] (49)

Thus |arg 119911| lt (12)120587Ω 120590 minus 120583120590 gt 0 120590 gt 0 0 lt 120583 lt 1 andR(120588 + 1) gt 06 Integrals Involving Multivariable Aleph-Function and General Class of Polynomials

The general class of polynomials 1198781198981 1198981199031198991 119899119903(119909) defined by

Srivastava [19] is as follows

1198781198981 1198981199031198991 119899119903(119909) = [11989911198981]sum

119897119894=0

sdot sdot sdot [119899119903119898119903]sum119897119903=0

119903prod119894=1

(minus119899119894)119898119894119897119894119897119894 119860119899119894 119897119894119909119897119894 (50)

Here 1198991 1198992 119899119903 isin N0 1198981 1198982 119898119903 isin N and thecoefficients 119860119899119894 119897119894

(119899119894 119897119894 ge 0) are any constant numbers Bysuitable restriction of the coefficient 119860119899119894 119897119894

the general classof polynomials has various special cases These include theJacobi polynomials the Laguerre polynomials the Besselpolynomials the Hermite polynomials the Gould-Hopperpolynomials and the Brafman polynomials ([20] p 158-161)

Next we establish the following integral

11986812 = int1

minus1(1 minus 119909)120588minus1 (1 + 119909)120590minus1 1198781198981 11989811990310158401198991 1198991199031015840

[119910 (1 minus 119909)120583 (1+ 119909)V] times alefsym0119899119880

119901119894119902119894 120591119894119877119881119882(1199111 (1 minus 119909)ℎ1

sdot (1 + 119909)1198961 119911119903 (1 minus 119909)ℎ119903 (1 + 119909)119896119903) 119889119909= [11989911198981]sum

1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895 119897119895119910119897119895 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 sdot sdot sdot 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

minus1(1 minus 119909)120588+sum119903119894=1 ℎ119894120585119894+120583sum1199031015840119895=1 119897119895minus1

sdot (1 + 119909)120590+sum119903119894=1 119896119894120585119894+Vsum1199031015840119895=1 119897119895minus1 1198891199091198891205851 119889120585119903

(51)

By applying (21) we have

11986812 = 2120588+120590minus1[11989911198981]sum1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895119897119895(2120583+V119910)119897119895 times 1(2120587120596)119903 int1198711 sdot sdot sdot

sdot int119871119903

120595 (1205851 sdot sdot sdot 120585119903) 119903prod119894=1

120601119894 (120585119894) (2ℎ119894+119896119894119911119894)120585119894 times Γ (120588 + sum119903119894=1 ℎ119894120585119894 + 120583sum1199031015840

119895=1 119897119895) Γ (120590 + sum119903119894=1 119896119894120585119894 + Vsum1199031015840

119895=1 119897119895)Γ (120588 + 120590 + sum119903

119894=1 (ℎ119894 + 119896119894) 120585119894 + (120583 + V)sum1199031015840

119895=1 119897119895) 1198891205851 119889120585119903(52)

11986812 = 2120588+120590minus1[11989911198981]sum1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895119897119895(2120583+V119910)119897119895

sdot alefsym0119899+1119880119901119894+2119902119894 120591119894119877119881119882

[[[[[[

2ℎ1+119896111991112ℎ119903+119896119903119911119903

| (1 minus 120588 minus 120583 1199031015840sum119895=1

119897119895 ℎ1 ℎ119903) (1 minus 120590 minus V1199031015840sum119895=1

119897119895 1198961 119896119903) 119860 119861(1 minus 120588 minus 120590 minus (120583 + V) 1199031015840sum

119895=1119897119895 (ℎ1 + 1198961) (ℎ119903 + 119896119903)) 119862 119863

]]]]]]

(53)

Journal of Applied Mathematics 9

These converge under the following conditions

(1) |arg 119911| lt (12)120587Ω(2) 120588 ge 1 120590 ge 1 120583 ge 0 V ge 0 ℎ ge 0 119896 ge 0 (ℎ and 119896 are

not both zero simultaneously)(3) R(120588) + ℎmin[R(119887119895119861119895)] gt 0 and R(120590) +119896min[R(119887119895119861119895)] gt 0

7 Special Cases

(i)By setting 120575 by 120578minus1 and120588 = 120590 = 120583 = V = 0 (32) transformsto the following integral

11986813 = 2120578 infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 119898)119896 (1 + 119899)119896Γ (1 + 119896) Γ (1 + 119896) (119896)2

times alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

211989711991112119897119911119903

| (1 minus 120578 minus 2119896 1198971 119897119903) 119860 119861(minus120578 minus 2119896 1198971 119897119903) 119862 119863

]]]]](54)

Thus |arg 119911| lt (12)120587Ω(ii)When we substitute 120588 by 120588 minus 1 and 120590 by 120590 minus 1 and put120583 = V = 0 integral (35) transforms to the following

11986814 = 2120588+120590minus1 infinsum119896=0

(minus119899)119896 (1 + 119899)119896Γ (1 + 119896) 119896 times alefsym0119899+2119880119901119894+2119902119894+1120591119894 119877119881119882

[[[[[

21198971+ℎ111991112119897119903+ℎ119903119911119903

| (1 minus 120588 minus 119896 1198971 119897119903) (1 minus 120590 ℎ1 ℎ119903) 119860 119861(1 minus 120588 minus 120590 minus 119896 (1198971 + ℎ1) (119897119903 + ℎ119903)) 119862 119863

]]]]] (55)

Thus |arg 119911| lt (12)120587Ω8 Concluding Remarks

In the present paper we evaluated new integrals involving themultivariable Aleph-function with certain special functionsCertain special cases of integrals that follow Remarks 1ndash3have been investigated in the literature by a number ofauthors [21ndash24]) with different arguments Therefore theresults presented in this paper are easily converted in termsof a similar type of new interesting integrals with differentarguments after some suitable parameter substitutions Inthis sequel one can obtain integral representation of moregeneralized special function which has many applications inphysics and engineering science

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C K Sharma and S S Ahmad ldquoOn the multivariable I-functionrdquo Acta Ciencia Indica Mathematics vol 20 no 2 pp113ndash116 1994

[2] Y N Prasad ldquoMultivariable I-functionrdquo Vijnana ParishadAnusandhan Patrika vol 29 pp 231ndash235 1986

[3] C K Sharma and G K Mishra ldquoExponential Fourier series forthe multivariable I-functionrdquo Acta Ciencia Indica Mathematicsvol 21 no 4 pp 495ndash501 1995

[4] H M Srivastava and R Panda ldquoSome expansion theoremsand generating relations for the H function of several complex

variablesrdquo Commentarii Mathematici Universitatis Sancti Paulivol 24 no 2 pp 119ndash137 1975

[5] N Sudland B Baumann andT FNonnenmacher ldquoOpen prob-lem Who knows about the Aleph () -functions rdquo FractionalCalculus and Applied Analysis vol 1 no 4 pp 401-402 1998

[6] D Kumar S D Purohit and J Choi ldquoGeneralized fractionalintegrals involving product of multivariable H-function and ageneral class of polynomialsrdquo Journal of Nonlinear Sciences andApplications vol 09 no 01 pp 8ndash21 2016

[7] R K Saxena and T K Pogany ldquoOn fractional integrationformulae for Aleph functionsrdquo Applied Mathematics and Com-putation vol 218 no 3 pp 985ndash990 2011

[8] R K Saxena J Ram and D Kumar ldquoGeneralized fractionalintegral of the product of two Aleph-functionsrdquo Applicationsand Applied Mathematics An International Journal (AAM) vol8 no 2 pp 631ndash646 2013

[9] R K Saxena J Ram and D L Suthar ldquoUnified fractionalderivative formulas for the multivariable H-functionrdquo VijnanaParishad Anusandhan Patrika vol 49 no 2 pp 159ndash175 2006

[10] D L Suthar and P Agarwal ldquoGeneralized mittag -leffler func-tion and the multivariable h-function involving the generalizedmellin-barnes contour integralsrdquoCommunications inNumericalAnalysis vol 2017 no 1 pp 25ndash33 2017

[11] D L Suthar B Debalkie and M Andualem ldquoModifiedSaigo fractional integral operators involving multivariableH-function and general class of multivariable polynomialsrdquoAdvances in Difference Equations vol 2019 no 1 article no 2132019

[12] D L Suthar H Habenom and H Tadesse ldquoCertain integralsinvolving aleph function and Wrightrsquos generalized hypergeo-metric functionrdquo Acta Universitatis Apulensis no 52 pp 1ndash102017

[13] D L Suthar G Reddy and B Shimelis ldquoOn Certain UnifiedFractional Integrals Pertaining to Product of Srivastavarsquos Poly-nomials and N-Functionrdquo International Journal of MathematicsTrends and Technology vol 47 no 1 pp 66ndash73 2017

[14] E D Rainville Special Functions The Macmillan CompanyNew York NY USA 1960

10 Journal of Applied Mathematics

[15] G Szego Orthogonal Polynomials American MathematicalSociety Colloquium Publications vol 23 American Mathemati-cal Society Providence RI USA 1959

[16] A Erdersquolyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol I McGraw-Hill NewYork-Toronto-London 1953

[17] V S Kiryakova Generalized Fractional Calculus and Applica-tions vol 301 of Pitman Research Notes in Mathematics UnitedStates with JohnWiley amp Sons Longman Scientific ampTechnicalNew York NY USA 1994

[18] V P Saxena ldquoFormal solution of certain new pair of dualintegral equations involving H-functionsrdquo Proceedings of theNational Academy of Sciences vol A52 no 3 pp 366ndash375 1982

[19] H M Srivastava ldquoA multilinear generating function for theKonhauser sets of biorthogonal polynomials suggested by theLaguerre polynomialsrdquo Pacific Journal of Mathematics vol 117no 1 pp 183ndash191 1985

[20] H M Srivastava and N P Singh ldquoThe integration of certainproducts of the multivariableH-function with a general class ofpolynomialsrdquo Rendiconti del Circolo Matematico di Palermo vol 32 no 2 pp 157ndash187 1983

[21] F Ayant ldquoThe integration of certains products of special func-tions and multivariable Aleph-functionsrdquo International Journalof Mathematics Trends and Technology vol 31 no 2 pp 101ndash1122016

[22] D Kumar R K Saxena and J Ram ldquoFinite integral formulasinvolving aleph functionrdquo Boletim da Sociedade Paranaense deMatematica vol 36 no 1 pp 177ndash193 2018

[23] D L Suthar S Agarwal and D Kumar ldquoCertain integralsinvolving the product of Gaussian hypergeometric function andAleph functionrdquoHonamMathematical Journal vol 41 no 1 pp1ndash17 2019

[24] D L Suthar and B Debalkie ldquoA new class of integral relationinvolving Aleph-functionsrdquo Surveys in Mathematics and itsApplications vol 12 pp 193ndash201 2017

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Submit your manuscripts atwwwhindawicom

4 Journal of Applied Mathematics

3 Integrals Involving Multivariable Aleph-Function with Jacobi Polynomials

The Jacobi polynomial ([15] 4212)) with parameters 120572 120573 isinR is defined by

119901(120572120573)119899 (119909)= (1 + 120572)119899119899 21198651 [minus119899 1 + 120572 + 120573 + 119899 1 + 120572 1 minus 1199092 ]

119899 ge 1(24)

where 21198651[] is the classical hypergeometric function Bysubstituting 120572 = 120573 = 0 the Jacobi polynomial (24) reducesto Lagrange polynomial ( [14] p 157) as

119901(120572120573)119899 (1) = (1 + 120572)119899119899 (25)

In this section we derive integral formulas involving multi-variable Aleph-functions multiplied by Jacobi polynomials

1198686= int1

minus1119909120582 (1

minus 119909)120572 (1 + 119909)120583 119875(120572120573)119899 (119909)alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[1199111 (1 + 119909)1198971

119911119903 (1 + 119909)119897119903] 119889119909

= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894)

sdot 119911120585119894119894 times int1

minus1119909120582 (1 minus 119909)120572 (1 + 119909)120583+sum119903119894=1 119897119894120585119894

sdot 119875(120572120573)119899 (119909) 119889119909 1198891205851 119889120585119903

(26)

Next we use the following formula

int1

minus1119909120582 (1 minus 119909)120572 (1 + 119909)120583 119901(120572120573)

119899 (119909) 119889119909= (minus1)119899 2120572+120583+1Γ (120583 + 1) Γ (119899 + 120572 + 1) Γ (120583 + 120573 + 1)

119899Γ (120583 + 120573 + 119899 + 1) Γ (120583 + 120572 + 119899 + 2)times 31198652 [ minus120582 120583 + 120573 + 1 120583 + 1

120583 + 120573 + 119899 + 1 120583 + 120572 + 119899 + 2 1] (27)

where 120572 gt minus1 and 120573 gt minus1 Also 31198652 is the special case ofgeneralized hypergeometric series

Then we have the following

= 1(2120587120596)119903sdot int

1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) rprod119894=1

120601119894 (120585119894) 119911120585119894119894 (minus1)119899 2120572+120583+sum119903119894=1 119897119894120585119894+1119899times Γ (120583 + sum119903

119894=1 119897119894120585119894 + 1) Γ (119899 + 120572 + 1) Γ (120583 + sum119903119894=1 119897119894120585119894 + 120573 + 1)

Γ (120583 + sum119903119894=1 119897119894120585119894 + 120573 + 119899 + 1) Γ (120583 + sum119903

119894=1 119897119894120585119894 + 120572 + 119899 + 2)

times 31198652

[[[[

minus120582 120583 + 119903sum119894=1

119897119894120585119894 + 120573 + 1 120583 + 119903sum119894=1

119897119894120585119894 + 1120583 + 119903sum

119894=1119897119894120585119894 + 120573 + 119899 + 1 120583 + 119903sum

119894=1119897119894120585119894 + 120572 + 119899 + 2 1]]]

]1198891205851

119889120585119903

(28)

Using the definition of hypergeometric function and somesimplifications the above expression becomes

1198686 = (minus1)119899 2120572+120583+1Γ (120572 + 119899 + 1)119899infinsum119896=0

(minus120582)119896 1119896119896 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) times (2119897119894119911119894)120585119894 Γ (120583 + sum119903119894=1 119897119894120585119894 + 120573 + 119896 + 1) Γ (120583 + sum119903

119894=1 119897119894120585119894 + 119896 + 1)Γ (120583 + sum119903

119894=1 119897119894120585119894 + 120573 + 119899 + 119896 + 1) Γ (120583 + sum119903119894=1 119897119894120585119894 + 120572 + 119899 + 119896 + 2)1198891205851

119889120585119903 = (minus1)119899 2120572+120583+1Γ (120572 + 119899 + 1)119899infinsum119896=0

(minus120582)119896 1119896119896

sdot alefsym0119899+2119880119901119894+2119902119894+2120591119894 119877119881119882

[[[[[

2119897111991112119897119903119911119903

| (minus120583 minus 120573 minus 119896 1198971 119897119903) (minus120583 minus 119896 1198971 119897119903) 119860 119861(minus120583 minus 120573 minus 119899 minus 119896 1198971 119897119903) (minus1 minus 120583 minus 120572 minus 119899 minus 119896 1198971 119897119903) 119862 119863

]]]]]

(29)

Journal of Applied Mathematics 5

Thus 120572 gt minus1 120573 gt minus1R(120582) gt minus1 and |arg 119911| lt (12)120587Ω

1198687= int1

minus1(1

minus 119909)120575 (1 + 119909)V 119875(120583V)119899 (119909) 119875(120588120590)

119898 (119909) times alefsym0119899119880119901119894119902119894 120591119894119877119881119882

[1199111 (1minus 119909)1198971 119911119903 (1 minus 119909)119897119903] 119889119909= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) 119911120585119894119894 times int1

minus1(1

minus 119909)120575+sum119903119894=1 119897119894120585119894 (1 + 119909)V 119875(120583V)119899 (119909)

sdot 119875(120588120590)119898 (119909) 1198891199091198891205851 119889120585119903

= Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899)119898119899sdot infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) 22119896 (119896)2times 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) 119911120585119894119894 times int1

minus1(1

minus 119909)1+120575+sum119903119894=1 119897119894120585119894+2119896minus1 (1 + 119909)1+Vminus1 1198891199091198891205851 119889120585119903

(30)

Now using (21) we have the following

1198687= Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899)119898119899sdot infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) 22119896 (119896)2times 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903)sdot 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 2120575+sum119903119894=1 119897119894120585119894+2119896+V+1 times 119861(120575 + 119903sum119894=1

119897119894120585119894 + 2119896

+ 1 V + 1)1198891205851 119889120585119903 = 2120575+V+1

sdot Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899) Γ (V + 1)119898119899

times infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) (119896)2

times 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903)sdot 119903prod119894=1

120601119894 (120585119894) (2119897119894119911119894)120585119894 Γ (120575 + sum119903119894=1 119897119894120585119894 + 2119896 + 1)

Γ (120575 + sum119903119894=1 119897119894120585119894 + 2119896 + V + 2)1198891205851

119889120585119903

(31)

Finally we arrive at

1198687 = 2120575+V+1 Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899) Γ (V + 1)119898119899 times infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) (119896)2

times alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

2119897111991112119897119903119911119903

| (minus120575 minus 2119896 1198971 119897119903) 119860 119861(minus1 minus 120575 minus 2119896 minus V 1198971 119897119903) 119862 119863

]]]]]

(32)

Thus 120575 gt 0R(V) gt minus1 and |arg 119911| lt (12)120587Ω

1198688 = int1

minus1(1 minus 119909)120588 (1 + 119909)120590 119875(120583V)

119899 (119909)

times alefsym0119899119880119901119894119902119894 120591119894119877119881119882

(1199111 (1 minus 119909)1198971 (1 + 119909)ℎ1 119911119903 (1

minus 119909)119897119903 (1 + 119909)ℎ119903) 119889119909 = Γ (1 + 120583 + 119899)119899

sdot infinsum119896=0

(minus119899)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120583 + 119896) 2119896119896 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851

120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 times int1

minus1(1 minus 119909)1+120588+sum119903119894=1 119897119894120585119894+119896minus1

sdot (1 + 119909)1+120590+sum119903119894=1 ℎ119894120585119894minus1 1198891199091198891205851 119889120585119903(33)

Using (21) we have the following

6 Journal of Applied Mathematics

1198688 = 2120588+120590+1 Γ (1 + 120583 + 119899)119899

infinsum119896=0

(minus119899)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120583 + 119896) 119896times 1(2120587120596)119903 int1198711 sdot sdot sdotsdot int

119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (2119897119894+ℎ119894119911119894)120585119894

times Γ (1 + 120588 + 119896 + sum119903119894=1 119897119894120585119894) Γ (1 + 120590 + sum119903

119894=1 ℎ119894120585119894)Γ (2 + 120588 + 120590 + 119896 + sum119903119894=1 (119897119894 + ℎ119894) 120585119894) 1198891205851

119889120585119903(34)

Finally rewriting the above equation by virtue of (1) we arriveat

1198688 = 2120588+120590+1 Γ (1 + 120583 + 119899)119899

infinsum119896=0

(minus119899)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120583 + 119896) 119896

times alefsym0119899+2119880119901119894+2119902119894+1120591119894 119877119881119882

[[[[[

21198971+ℎ111991112119897119903+ℎ119903119911119903

| (minus120588 minus 119896 1198971 119897119903) (minus120590 ℎ1 ℎ119903) 119860 119861(minus1 minus 120588 minus 120590 minus 119896) (1198971 + ℎ1) (119897119903 + ℎ119903)) 119862 119863

]]]]]

(35)

ThusR(V) gt minus1R(120583) gt minus1 and |arg 119911| lt (12)120587Ω4 Integrals Involving Multivariable Aleph-Function with Legendre Function

The solution of Legendre differential equation in the form ofGauss hypergeometric type is as follows

119901120583V (119909) = 1Γ (1 minus 120583) (119909 + 1119909 minus 1)

12120583

sdot 21198651 [minusV V + 1 1 minus 120583 1 minus 1199092 ] |1 minus 119909| lt 2

(36)

Here119901120583V (119909) is known as the Legendre function of the first kind

[16]

Next we derive the integrals with Legendre function

1198689 = int1

0119909120590minus1 (1 minus 1199092)1205832 119901120583

V (119909)sdot alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205881 119911119903119909120588119903] 119889119909 = 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

0119909120590+sum119903119894=1 120588119894120585119894minus1 (1 minus 1199092)1205832 119875120583

V (119909) 1198891199091198891205851 119889120585119903

(37)

Next we use the following formula ([16] Sec 312) for 120583 isinNR(120590) gt 0

int1

0119909120590minus1 (1 minus 1199092)1205832 119901120583

V (119909) 119889119909 = (minus1)120583 2minus120590minus12058312058712Γ (120590) Γ (1 + 120583 + V)Γ (1 minus 120583 + V) Γ (12 + 1205902 + 1205832 minus V2) Γ (1 + 1205902 + 1205832 + V2) (38)

Now applying (38) we obtain

1198689 = 2minus120590minus120583 (minus1)120583 12058712 Γ (1 + 120583 + V)Γ (1 minus 120583 + V) 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) (2minus120588119894119911119894)120585119894

Journal of Applied Mathematics 7

times Γ (120590 + sum119903119894=1 120588119894120585119894)Γ (12 + 1205902 + 1205832 minus V2 + (sum119903

119894=1 120588119894120585119894) 2) Γ (1 + 1205902 + 1205832 + V2 + (sum119903119894=1 120588119894120585119894) 2)1198891205851 119889120585119903

(39)

1198689 = 2minus120590minus120583 (minus1)120583 12058712 Γ (1 + 120583 + V)Γ (1 minus 120583 + V)

sdot alefsym0119899+1119880119901119894+1119902119894+2120591119894 119877119881119882

[[[[[

2minus120588111991112minus120588119903119911119903

| (1 minus 120590 1205881 120588119903) 119860 119861(12 minus 1205832 + V2 minus 1205902 12058812 1205881199032 ) (minus1205832 minus V2 minus 1205902 12058812 1205881199032 ) 119862 119863

]]]]]

(40)

Thus |arg (119911)| lt (12)120587Ω 120590 gt 0 and 120583 isin N cup 011986810 = int1

0119909120590minus1 (1 minus 1199092)minus1205832 119901120583

V (119909)sdot alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205881 119911119903119909120588119903] 119889119909 = 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

0119909120590+sum119903119894=1 120588119894120585119894minus1 (1 minus 1199092)minus1205832 119875120583

V (119909) 1198891199091198891205851

119889120585119903(41)

Next we use the following formula ([16] Sec 312) for 120583 isinNR(120590) gt 0

int1

0119909120590minus1 (1 minus 1199092)minus1205832 119901120583

V (119909) 119889119909= 2minus120590+12058312058712Γ (120590)Γ (12 + 1205902 minus 1205832 minus V2) Γ (1 + 1205902 minus 1205832 minus V2)

(42)

Using (42) we obtain

11986810 = 2minus120590+12058312058712 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (2minus120588119894119911119894)120585119894

times Γ (120590 + sum119903119894=1 120588119894120585119894)Γ (12 + 1205902 minus 1205832 minus V2 + (sum119903

119894=1 120588119894120585119894) 2) Γ (1 + 1205902 minus 1205832 minus V2 + (sum119903119894=1 120588119894120585119894) 2)1198891205851 119889120585119903

(43)

11986810 = 2minus120590+12058312058712alefsym0119899+1119880119901119894+1119902119894+2120591119894119877119881119882

[[[[[

2minus120588111991112minus120588119903119911119903

| (1 minus 120590 1205881 120588119903) 119860 119861(12 + 1205832 + V2 minus 1205902 12058812 1205881199032 ) (1205832 + V2 minus 1205902 12058812 1205881199032 ) 119862 119863

]]]]] (44)

Thus |arg (119911)| lt (12)120587ΩR(120590) gt 0 andR(120583) isin N cup 0

5 Integrals Involving Multivariable Aleph-Function and Bessel-Maitland Function

The Bessel-Maitland function is defined by Kiryakova [17] asfollows

119869120583V (119909) = 120601 (120583 V + 1 119909) = infinsum119896=0

1Γ (120583119896 + V + 1) (minus119909)119896

119896 120583 gt minus1 119909 isin C

(45)

11986811 = intinfin

0119909120588119869120583V (119909)

sdot alefsym0119899119880119901119894119902119894120591119894 119877119881119882

[11991111199091205901 119911119903119909120590119903] 119889119909 119889119909= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585i) 119911120585119894119894

sdot intinfin

0119909120588+sum119903119894=1 120590119894120585119894119869120583V (119909) 119889119909 1198891205851 119889120585119903

(46)

Now using the following formula from [18]

8 Journal of Applied Mathematics

intinfin

0119909120588119869120583V (119909) 119889119909 = Γ (120588 + 1)

Γ (1 + V minus 120583 minus 120583120588)(R (120588) gt minus1 0 lt 120583 lt 1)

(47)and further applying (47) we obtain

11986814 = 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 Γ (1 + 120588 + sum119903119894=1 120590119894120585119894)Γ (1 + V minus 120583 minus 120583120588 minus 120583sum119903

119894=1 120590119894120585119894)1198891205851 119889120585119903 (48)

11986814 = alefsym0119899+1119880119901119894+2119902119894 120591119894119877119881119882

[[[[[

1199111119911119903

| (minus120588 1205901 120590119903) (1 + V minus 120583 minus 120583120588 1205831205901 120583120590119903) 119860 119861119862 119863

]]]]] (49)

Thus |arg 119911| lt (12)120587Ω 120590 minus 120583120590 gt 0 120590 gt 0 0 lt 120583 lt 1 andR(120588 + 1) gt 06 Integrals Involving Multivariable Aleph-Function and General Class of Polynomials

The general class of polynomials 1198781198981 1198981199031198991 119899119903(119909) defined by

Srivastava [19] is as follows

1198781198981 1198981199031198991 119899119903(119909) = [11989911198981]sum

119897119894=0

sdot sdot sdot [119899119903119898119903]sum119897119903=0

119903prod119894=1

(minus119899119894)119898119894119897119894119897119894 119860119899119894 119897119894119909119897119894 (50)

Here 1198991 1198992 119899119903 isin N0 1198981 1198982 119898119903 isin N and thecoefficients 119860119899119894 119897119894

(119899119894 119897119894 ge 0) are any constant numbers Bysuitable restriction of the coefficient 119860119899119894 119897119894

the general classof polynomials has various special cases These include theJacobi polynomials the Laguerre polynomials the Besselpolynomials the Hermite polynomials the Gould-Hopperpolynomials and the Brafman polynomials ([20] p 158-161)

Next we establish the following integral

11986812 = int1

minus1(1 minus 119909)120588minus1 (1 + 119909)120590minus1 1198781198981 11989811990310158401198991 1198991199031015840

[119910 (1 minus 119909)120583 (1+ 119909)V] times alefsym0119899119880

119901119894119902119894 120591119894119877119881119882(1199111 (1 minus 119909)ℎ1

sdot (1 + 119909)1198961 119911119903 (1 minus 119909)ℎ119903 (1 + 119909)119896119903) 119889119909= [11989911198981]sum

1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895 119897119895119910119897119895 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 sdot sdot sdot 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

minus1(1 minus 119909)120588+sum119903119894=1 ℎ119894120585119894+120583sum1199031015840119895=1 119897119895minus1

sdot (1 + 119909)120590+sum119903119894=1 119896119894120585119894+Vsum1199031015840119895=1 119897119895minus1 1198891199091198891205851 119889120585119903

(51)

By applying (21) we have

11986812 = 2120588+120590minus1[11989911198981]sum1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895119897119895(2120583+V119910)119897119895 times 1(2120587120596)119903 int1198711 sdot sdot sdot

sdot int119871119903

120595 (1205851 sdot sdot sdot 120585119903) 119903prod119894=1

120601119894 (120585119894) (2ℎ119894+119896119894119911119894)120585119894 times Γ (120588 + sum119903119894=1 ℎ119894120585119894 + 120583sum1199031015840

119895=1 119897119895) Γ (120590 + sum119903119894=1 119896119894120585119894 + Vsum1199031015840

119895=1 119897119895)Γ (120588 + 120590 + sum119903

119894=1 (ℎ119894 + 119896119894) 120585119894 + (120583 + V)sum1199031015840

119895=1 119897119895) 1198891205851 119889120585119903(52)

11986812 = 2120588+120590minus1[11989911198981]sum1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895119897119895(2120583+V119910)119897119895

sdot alefsym0119899+1119880119901119894+2119902119894 120591119894119877119881119882

[[[[[[

2ℎ1+119896111991112ℎ119903+119896119903119911119903

| (1 minus 120588 minus 120583 1199031015840sum119895=1

119897119895 ℎ1 ℎ119903) (1 minus 120590 minus V1199031015840sum119895=1

119897119895 1198961 119896119903) 119860 119861(1 minus 120588 minus 120590 minus (120583 + V) 1199031015840sum

119895=1119897119895 (ℎ1 + 1198961) (ℎ119903 + 119896119903)) 119862 119863

]]]]]]

(53)

Journal of Applied Mathematics 9

These converge under the following conditions

(1) |arg 119911| lt (12)120587Ω(2) 120588 ge 1 120590 ge 1 120583 ge 0 V ge 0 ℎ ge 0 119896 ge 0 (ℎ and 119896 are

not both zero simultaneously)(3) R(120588) + ℎmin[R(119887119895119861119895)] gt 0 and R(120590) +119896min[R(119887119895119861119895)] gt 0

7 Special Cases

(i)By setting 120575 by 120578minus1 and120588 = 120590 = 120583 = V = 0 (32) transformsto the following integral

11986813 = 2120578 infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 119898)119896 (1 + 119899)119896Γ (1 + 119896) Γ (1 + 119896) (119896)2

times alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

211989711991112119897119911119903

| (1 minus 120578 minus 2119896 1198971 119897119903) 119860 119861(minus120578 minus 2119896 1198971 119897119903) 119862 119863

]]]]](54)

Thus |arg 119911| lt (12)120587Ω(ii)When we substitute 120588 by 120588 minus 1 and 120590 by 120590 minus 1 and put120583 = V = 0 integral (35) transforms to the following

11986814 = 2120588+120590minus1 infinsum119896=0

(minus119899)119896 (1 + 119899)119896Γ (1 + 119896) 119896 times alefsym0119899+2119880119901119894+2119902119894+1120591119894 119877119881119882

[[[[[

21198971+ℎ111991112119897119903+ℎ119903119911119903

| (1 minus 120588 minus 119896 1198971 119897119903) (1 minus 120590 ℎ1 ℎ119903) 119860 119861(1 minus 120588 minus 120590 minus 119896 (1198971 + ℎ1) (119897119903 + ℎ119903)) 119862 119863

]]]]] (55)

Thus |arg 119911| lt (12)120587Ω8 Concluding Remarks

In the present paper we evaluated new integrals involving themultivariable Aleph-function with certain special functionsCertain special cases of integrals that follow Remarks 1ndash3have been investigated in the literature by a number ofauthors [21ndash24]) with different arguments Therefore theresults presented in this paper are easily converted in termsof a similar type of new interesting integrals with differentarguments after some suitable parameter substitutions Inthis sequel one can obtain integral representation of moregeneralized special function which has many applications inphysics and engineering science

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C K Sharma and S S Ahmad ldquoOn the multivariable I-functionrdquo Acta Ciencia Indica Mathematics vol 20 no 2 pp113ndash116 1994

[2] Y N Prasad ldquoMultivariable I-functionrdquo Vijnana ParishadAnusandhan Patrika vol 29 pp 231ndash235 1986

[3] C K Sharma and G K Mishra ldquoExponential Fourier series forthe multivariable I-functionrdquo Acta Ciencia Indica Mathematicsvol 21 no 4 pp 495ndash501 1995

[4] H M Srivastava and R Panda ldquoSome expansion theoremsand generating relations for the H function of several complex

variablesrdquo Commentarii Mathematici Universitatis Sancti Paulivol 24 no 2 pp 119ndash137 1975

[5] N Sudland B Baumann andT FNonnenmacher ldquoOpen prob-lem Who knows about the Aleph () -functions rdquo FractionalCalculus and Applied Analysis vol 1 no 4 pp 401-402 1998

[6] D Kumar S D Purohit and J Choi ldquoGeneralized fractionalintegrals involving product of multivariable H-function and ageneral class of polynomialsrdquo Journal of Nonlinear Sciences andApplications vol 09 no 01 pp 8ndash21 2016

[7] R K Saxena and T K Pogany ldquoOn fractional integrationformulae for Aleph functionsrdquo Applied Mathematics and Com-putation vol 218 no 3 pp 985ndash990 2011

[8] R K Saxena J Ram and D Kumar ldquoGeneralized fractionalintegral of the product of two Aleph-functionsrdquo Applicationsand Applied Mathematics An International Journal (AAM) vol8 no 2 pp 631ndash646 2013

[9] R K Saxena J Ram and D L Suthar ldquoUnified fractionalderivative formulas for the multivariable H-functionrdquo VijnanaParishad Anusandhan Patrika vol 49 no 2 pp 159ndash175 2006

[10] D L Suthar and P Agarwal ldquoGeneralized mittag -leffler func-tion and the multivariable h-function involving the generalizedmellin-barnes contour integralsrdquoCommunications inNumericalAnalysis vol 2017 no 1 pp 25ndash33 2017

[11] D L Suthar B Debalkie and M Andualem ldquoModifiedSaigo fractional integral operators involving multivariableH-function and general class of multivariable polynomialsrdquoAdvances in Difference Equations vol 2019 no 1 article no 2132019

[12] D L Suthar H Habenom and H Tadesse ldquoCertain integralsinvolving aleph function and Wrightrsquos generalized hypergeo-metric functionrdquo Acta Universitatis Apulensis no 52 pp 1ndash102017

[13] D L Suthar G Reddy and B Shimelis ldquoOn Certain UnifiedFractional Integrals Pertaining to Product of Srivastavarsquos Poly-nomials and N-Functionrdquo International Journal of MathematicsTrends and Technology vol 47 no 1 pp 66ndash73 2017

[14] E D Rainville Special Functions The Macmillan CompanyNew York NY USA 1960

10 Journal of Applied Mathematics

[15] G Szego Orthogonal Polynomials American MathematicalSociety Colloquium Publications vol 23 American Mathemati-cal Society Providence RI USA 1959

[16] A Erdersquolyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol I McGraw-Hill NewYork-Toronto-London 1953

[17] V S Kiryakova Generalized Fractional Calculus and Applica-tions vol 301 of Pitman Research Notes in Mathematics UnitedStates with JohnWiley amp Sons Longman Scientific ampTechnicalNew York NY USA 1994

[18] V P Saxena ldquoFormal solution of certain new pair of dualintegral equations involving H-functionsrdquo Proceedings of theNational Academy of Sciences vol A52 no 3 pp 366ndash375 1982

[19] H M Srivastava ldquoA multilinear generating function for theKonhauser sets of biorthogonal polynomials suggested by theLaguerre polynomialsrdquo Pacific Journal of Mathematics vol 117no 1 pp 183ndash191 1985

[20] H M Srivastava and N P Singh ldquoThe integration of certainproducts of the multivariableH-function with a general class ofpolynomialsrdquo Rendiconti del Circolo Matematico di Palermo vol 32 no 2 pp 157ndash187 1983

[21] F Ayant ldquoThe integration of certains products of special func-tions and multivariable Aleph-functionsrdquo International Journalof Mathematics Trends and Technology vol 31 no 2 pp 101ndash1122016

[22] D Kumar R K Saxena and J Ram ldquoFinite integral formulasinvolving aleph functionrdquo Boletim da Sociedade Paranaense deMatematica vol 36 no 1 pp 177ndash193 2018

[23] D L Suthar S Agarwal and D Kumar ldquoCertain integralsinvolving the product of Gaussian hypergeometric function andAleph functionrdquoHonamMathematical Journal vol 41 no 1 pp1ndash17 2019

[24] D L Suthar and B Debalkie ldquoA new class of integral relationinvolving Aleph-functionsrdquo Surveys in Mathematics and itsApplications vol 12 pp 193ndash201 2017

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Journal of Applied Mathematics 5

Thus 120572 gt minus1 120573 gt minus1R(120582) gt minus1 and |arg 119911| lt (12)120587Ω

1198687= int1

minus1(1

minus 119909)120575 (1 + 119909)V 119875(120583V)119899 (119909) 119875(120588120590)

119898 (119909) times alefsym0119899119880119901119894119902119894 120591119894119877119881119882

[1199111 (1minus 119909)1198971 119911119903 (1 minus 119909)119897119903] 119889119909= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) 119911120585119894119894 times int1

minus1(1

minus 119909)120575+sum119903119894=1 119897119894120585119894 (1 + 119909)V 119875(120583V)119899 (119909)

sdot 119875(120588120590)119898 (119909) 1198891199091198891205851 119889120585119903

= Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899)119898119899sdot infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) 22119896 (119896)2times 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) 119911120585119894119894 times int1

minus1(1

minus 119909)1+120575+sum119903119894=1 119897119894120585119894+2119896minus1 (1 + 119909)1+Vminus1 1198891199091198891205851 119889120585119903

(30)

Now using (21) we have the following

1198687= Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899)119898119899sdot infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) 22119896 (119896)2times 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903)sdot 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 2120575+sum119903119894=1 119897119894120585119894+2119896+V+1 times 119861(120575 + 119903sum119894=1

119897119894120585119894 + 2119896

+ 1 V + 1)1198891205851 119889120585119903 = 2120575+V+1

sdot Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899) Γ (V + 1)119898119899

times infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) (119896)2

times 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903)sdot 119903prod119894=1

120601119894 (120585119894) (2119897119894119911119894)120585119894 Γ (120575 + sum119903119894=1 119897119894120585119894 + 2119896 + 1)

Γ (120575 + sum119903119894=1 119897119894120585119894 + 2119896 + V + 2)1198891205851

119889120585119903

(31)

Finally we arrive at

1198687 = 2120575+V+1 Γ (1 + 120588 + 119898) Γ (1 + 120583 + 119899) Γ (V + 1)119898119899 times infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 120588 + 120590 + 119898)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120588 + 119896) Γ (1 + 120583 + 119896) (119896)2

times alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

2119897111991112119897119903119911119903

| (minus120575 minus 2119896 1198971 119897119903) 119860 119861(minus1 minus 120575 minus 2119896 minus V 1198971 119897119903) 119862 119863

]]]]]

(32)

Thus 120575 gt 0R(V) gt minus1 and |arg 119911| lt (12)120587Ω

1198688 = int1

minus1(1 minus 119909)120588 (1 + 119909)120590 119875(120583V)

119899 (119909)

times alefsym0119899119880119901119894119902119894 120591119894119877119881119882

(1199111 (1 minus 119909)1198971 (1 + 119909)ℎ1 119911119903 (1

minus 119909)119897119903 (1 + 119909)ℎ119903) 119889119909 = Γ (1 + 120583 + 119899)119899

sdot infinsum119896=0

(minus119899)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120583 + 119896) 2119896119896 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851

120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 times int1

minus1(1 minus 119909)1+120588+sum119903119894=1 119897119894120585119894+119896minus1

sdot (1 + 119909)1+120590+sum119903119894=1 ℎ119894120585119894minus1 1198891199091198891205851 119889120585119903(33)

Using (21) we have the following

6 Journal of Applied Mathematics

1198688 = 2120588+120590+1 Γ (1 + 120583 + 119899)119899

infinsum119896=0

(minus119899)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120583 + 119896) 119896times 1(2120587120596)119903 int1198711 sdot sdot sdotsdot int

119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (2119897119894+ℎ119894119911119894)120585119894

times Γ (1 + 120588 + 119896 + sum119903119894=1 119897119894120585119894) Γ (1 + 120590 + sum119903

119894=1 ℎ119894120585119894)Γ (2 + 120588 + 120590 + 119896 + sum119903119894=1 (119897119894 + ℎ119894) 120585119894) 1198891205851

119889120585119903(34)

Finally rewriting the above equation by virtue of (1) we arriveat

1198688 = 2120588+120590+1 Γ (1 + 120583 + 119899)119899

infinsum119896=0

(minus119899)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120583 + 119896) 119896

times alefsym0119899+2119880119901119894+2119902119894+1120591119894 119877119881119882

[[[[[

21198971+ℎ111991112119897119903+ℎ119903119911119903

| (minus120588 minus 119896 1198971 119897119903) (minus120590 ℎ1 ℎ119903) 119860 119861(minus1 minus 120588 minus 120590 minus 119896) (1198971 + ℎ1) (119897119903 + ℎ119903)) 119862 119863

]]]]]

(35)

ThusR(V) gt minus1R(120583) gt minus1 and |arg 119911| lt (12)120587Ω4 Integrals Involving Multivariable Aleph-Function with Legendre Function

The solution of Legendre differential equation in the form ofGauss hypergeometric type is as follows

119901120583V (119909) = 1Γ (1 minus 120583) (119909 + 1119909 minus 1)

12120583

sdot 21198651 [minusV V + 1 1 minus 120583 1 minus 1199092 ] |1 minus 119909| lt 2

(36)

Here119901120583V (119909) is known as the Legendre function of the first kind

[16]

Next we derive the integrals with Legendre function

1198689 = int1

0119909120590minus1 (1 minus 1199092)1205832 119901120583

V (119909)sdot alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205881 119911119903119909120588119903] 119889119909 = 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

0119909120590+sum119903119894=1 120588119894120585119894minus1 (1 minus 1199092)1205832 119875120583

V (119909) 1198891199091198891205851 119889120585119903

(37)

Next we use the following formula ([16] Sec 312) for 120583 isinNR(120590) gt 0

int1

0119909120590minus1 (1 minus 1199092)1205832 119901120583

V (119909) 119889119909 = (minus1)120583 2minus120590minus12058312058712Γ (120590) Γ (1 + 120583 + V)Γ (1 minus 120583 + V) Γ (12 + 1205902 + 1205832 minus V2) Γ (1 + 1205902 + 1205832 + V2) (38)

Now applying (38) we obtain

1198689 = 2minus120590minus120583 (minus1)120583 12058712 Γ (1 + 120583 + V)Γ (1 minus 120583 + V) 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) (2minus120588119894119911119894)120585119894

Journal of Applied Mathematics 7

times Γ (120590 + sum119903119894=1 120588119894120585119894)Γ (12 + 1205902 + 1205832 minus V2 + (sum119903

119894=1 120588119894120585119894) 2) Γ (1 + 1205902 + 1205832 + V2 + (sum119903119894=1 120588119894120585119894) 2)1198891205851 119889120585119903

(39)

1198689 = 2minus120590minus120583 (minus1)120583 12058712 Γ (1 + 120583 + V)Γ (1 minus 120583 + V)

sdot alefsym0119899+1119880119901119894+1119902119894+2120591119894 119877119881119882

[[[[[

2minus120588111991112minus120588119903119911119903

| (1 minus 120590 1205881 120588119903) 119860 119861(12 minus 1205832 + V2 minus 1205902 12058812 1205881199032 ) (minus1205832 minus V2 minus 1205902 12058812 1205881199032 ) 119862 119863

]]]]]

(40)

Thus |arg (119911)| lt (12)120587Ω 120590 gt 0 and 120583 isin N cup 011986810 = int1

0119909120590minus1 (1 minus 1199092)minus1205832 119901120583

V (119909)sdot alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205881 119911119903119909120588119903] 119889119909 = 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

0119909120590+sum119903119894=1 120588119894120585119894minus1 (1 minus 1199092)minus1205832 119875120583

V (119909) 1198891199091198891205851

119889120585119903(41)

Next we use the following formula ([16] Sec 312) for 120583 isinNR(120590) gt 0

int1

0119909120590minus1 (1 minus 1199092)minus1205832 119901120583

V (119909) 119889119909= 2minus120590+12058312058712Γ (120590)Γ (12 + 1205902 minus 1205832 minus V2) Γ (1 + 1205902 minus 1205832 minus V2)

(42)

Using (42) we obtain

11986810 = 2minus120590+12058312058712 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (2minus120588119894119911119894)120585119894

times Γ (120590 + sum119903119894=1 120588119894120585119894)Γ (12 + 1205902 minus 1205832 minus V2 + (sum119903

119894=1 120588119894120585119894) 2) Γ (1 + 1205902 minus 1205832 minus V2 + (sum119903119894=1 120588119894120585119894) 2)1198891205851 119889120585119903

(43)

11986810 = 2minus120590+12058312058712alefsym0119899+1119880119901119894+1119902119894+2120591119894119877119881119882

[[[[[

2minus120588111991112minus120588119903119911119903

| (1 minus 120590 1205881 120588119903) 119860 119861(12 + 1205832 + V2 minus 1205902 12058812 1205881199032 ) (1205832 + V2 minus 1205902 12058812 1205881199032 ) 119862 119863

]]]]] (44)

Thus |arg (119911)| lt (12)120587ΩR(120590) gt 0 andR(120583) isin N cup 0

5 Integrals Involving Multivariable Aleph-Function and Bessel-Maitland Function

The Bessel-Maitland function is defined by Kiryakova [17] asfollows

119869120583V (119909) = 120601 (120583 V + 1 119909) = infinsum119896=0

1Γ (120583119896 + V + 1) (minus119909)119896

119896 120583 gt minus1 119909 isin C

(45)

11986811 = intinfin

0119909120588119869120583V (119909)

sdot alefsym0119899119880119901119894119902119894120591119894 119877119881119882

[11991111199091205901 119911119903119909120590119903] 119889119909 119889119909= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585i) 119911120585119894119894

sdot intinfin

0119909120588+sum119903119894=1 120590119894120585119894119869120583V (119909) 119889119909 1198891205851 119889120585119903

(46)

Now using the following formula from [18]

8 Journal of Applied Mathematics

intinfin

0119909120588119869120583V (119909) 119889119909 = Γ (120588 + 1)

Γ (1 + V minus 120583 minus 120583120588)(R (120588) gt minus1 0 lt 120583 lt 1)

(47)and further applying (47) we obtain

11986814 = 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 Γ (1 + 120588 + sum119903119894=1 120590119894120585119894)Γ (1 + V minus 120583 minus 120583120588 minus 120583sum119903

119894=1 120590119894120585119894)1198891205851 119889120585119903 (48)

11986814 = alefsym0119899+1119880119901119894+2119902119894 120591119894119877119881119882

[[[[[

1199111119911119903

| (minus120588 1205901 120590119903) (1 + V minus 120583 minus 120583120588 1205831205901 120583120590119903) 119860 119861119862 119863

]]]]] (49)

Thus |arg 119911| lt (12)120587Ω 120590 minus 120583120590 gt 0 120590 gt 0 0 lt 120583 lt 1 andR(120588 + 1) gt 06 Integrals Involving Multivariable Aleph-Function and General Class of Polynomials

The general class of polynomials 1198781198981 1198981199031198991 119899119903(119909) defined by

Srivastava [19] is as follows

1198781198981 1198981199031198991 119899119903(119909) = [11989911198981]sum

119897119894=0

sdot sdot sdot [119899119903119898119903]sum119897119903=0

119903prod119894=1

(minus119899119894)119898119894119897119894119897119894 119860119899119894 119897119894119909119897119894 (50)

Here 1198991 1198992 119899119903 isin N0 1198981 1198982 119898119903 isin N and thecoefficients 119860119899119894 119897119894

(119899119894 119897119894 ge 0) are any constant numbers Bysuitable restriction of the coefficient 119860119899119894 119897119894

the general classof polynomials has various special cases These include theJacobi polynomials the Laguerre polynomials the Besselpolynomials the Hermite polynomials the Gould-Hopperpolynomials and the Brafman polynomials ([20] p 158-161)

Next we establish the following integral

11986812 = int1

minus1(1 minus 119909)120588minus1 (1 + 119909)120590minus1 1198781198981 11989811990310158401198991 1198991199031015840

[119910 (1 minus 119909)120583 (1+ 119909)V] times alefsym0119899119880

119901119894119902119894 120591119894119877119881119882(1199111 (1 minus 119909)ℎ1

sdot (1 + 119909)1198961 119911119903 (1 minus 119909)ℎ119903 (1 + 119909)119896119903) 119889119909= [11989911198981]sum

1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895 119897119895119910119897119895 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 sdot sdot sdot 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

minus1(1 minus 119909)120588+sum119903119894=1 ℎ119894120585119894+120583sum1199031015840119895=1 119897119895minus1

sdot (1 + 119909)120590+sum119903119894=1 119896119894120585119894+Vsum1199031015840119895=1 119897119895minus1 1198891199091198891205851 119889120585119903

(51)

By applying (21) we have

11986812 = 2120588+120590minus1[11989911198981]sum1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895119897119895(2120583+V119910)119897119895 times 1(2120587120596)119903 int1198711 sdot sdot sdot

sdot int119871119903

120595 (1205851 sdot sdot sdot 120585119903) 119903prod119894=1

120601119894 (120585119894) (2ℎ119894+119896119894119911119894)120585119894 times Γ (120588 + sum119903119894=1 ℎ119894120585119894 + 120583sum1199031015840

119895=1 119897119895) Γ (120590 + sum119903119894=1 119896119894120585119894 + Vsum1199031015840

119895=1 119897119895)Γ (120588 + 120590 + sum119903

119894=1 (ℎ119894 + 119896119894) 120585119894 + (120583 + V)sum1199031015840

119895=1 119897119895) 1198891205851 119889120585119903(52)

11986812 = 2120588+120590minus1[11989911198981]sum1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895119897119895(2120583+V119910)119897119895

sdot alefsym0119899+1119880119901119894+2119902119894 120591119894119877119881119882

[[[[[[

2ℎ1+119896111991112ℎ119903+119896119903119911119903

| (1 minus 120588 minus 120583 1199031015840sum119895=1

119897119895 ℎ1 ℎ119903) (1 minus 120590 minus V1199031015840sum119895=1

119897119895 1198961 119896119903) 119860 119861(1 minus 120588 minus 120590 minus (120583 + V) 1199031015840sum

119895=1119897119895 (ℎ1 + 1198961) (ℎ119903 + 119896119903)) 119862 119863

]]]]]]

(53)

Journal of Applied Mathematics 9

These converge under the following conditions

(1) |arg 119911| lt (12)120587Ω(2) 120588 ge 1 120590 ge 1 120583 ge 0 V ge 0 ℎ ge 0 119896 ge 0 (ℎ and 119896 are

not both zero simultaneously)(3) R(120588) + ℎmin[R(119887119895119861119895)] gt 0 and R(120590) +119896min[R(119887119895119861119895)] gt 0

7 Special Cases

(i)By setting 120575 by 120578minus1 and120588 = 120590 = 120583 = V = 0 (32) transformsto the following integral

11986813 = 2120578 infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 119898)119896 (1 + 119899)119896Γ (1 + 119896) Γ (1 + 119896) (119896)2

times alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

211989711991112119897119911119903

| (1 minus 120578 minus 2119896 1198971 119897119903) 119860 119861(minus120578 minus 2119896 1198971 119897119903) 119862 119863

]]]]](54)

Thus |arg 119911| lt (12)120587Ω(ii)When we substitute 120588 by 120588 minus 1 and 120590 by 120590 minus 1 and put120583 = V = 0 integral (35) transforms to the following

11986814 = 2120588+120590minus1 infinsum119896=0

(minus119899)119896 (1 + 119899)119896Γ (1 + 119896) 119896 times alefsym0119899+2119880119901119894+2119902119894+1120591119894 119877119881119882

[[[[[

21198971+ℎ111991112119897119903+ℎ119903119911119903

| (1 minus 120588 minus 119896 1198971 119897119903) (1 minus 120590 ℎ1 ℎ119903) 119860 119861(1 minus 120588 minus 120590 minus 119896 (1198971 + ℎ1) (119897119903 + ℎ119903)) 119862 119863

]]]]] (55)

Thus |arg 119911| lt (12)120587Ω8 Concluding Remarks

In the present paper we evaluated new integrals involving themultivariable Aleph-function with certain special functionsCertain special cases of integrals that follow Remarks 1ndash3have been investigated in the literature by a number ofauthors [21ndash24]) with different arguments Therefore theresults presented in this paper are easily converted in termsof a similar type of new interesting integrals with differentarguments after some suitable parameter substitutions Inthis sequel one can obtain integral representation of moregeneralized special function which has many applications inphysics and engineering science

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C K Sharma and S S Ahmad ldquoOn the multivariable I-functionrdquo Acta Ciencia Indica Mathematics vol 20 no 2 pp113ndash116 1994

[2] Y N Prasad ldquoMultivariable I-functionrdquo Vijnana ParishadAnusandhan Patrika vol 29 pp 231ndash235 1986

[3] C K Sharma and G K Mishra ldquoExponential Fourier series forthe multivariable I-functionrdquo Acta Ciencia Indica Mathematicsvol 21 no 4 pp 495ndash501 1995

[4] H M Srivastava and R Panda ldquoSome expansion theoremsand generating relations for the H function of several complex

variablesrdquo Commentarii Mathematici Universitatis Sancti Paulivol 24 no 2 pp 119ndash137 1975

[5] N Sudland B Baumann andT FNonnenmacher ldquoOpen prob-lem Who knows about the Aleph () -functions rdquo FractionalCalculus and Applied Analysis vol 1 no 4 pp 401-402 1998

[6] D Kumar S D Purohit and J Choi ldquoGeneralized fractionalintegrals involving product of multivariable H-function and ageneral class of polynomialsrdquo Journal of Nonlinear Sciences andApplications vol 09 no 01 pp 8ndash21 2016

[7] R K Saxena and T K Pogany ldquoOn fractional integrationformulae for Aleph functionsrdquo Applied Mathematics and Com-putation vol 218 no 3 pp 985ndash990 2011

[8] R K Saxena J Ram and D Kumar ldquoGeneralized fractionalintegral of the product of two Aleph-functionsrdquo Applicationsand Applied Mathematics An International Journal (AAM) vol8 no 2 pp 631ndash646 2013

[9] R K Saxena J Ram and D L Suthar ldquoUnified fractionalderivative formulas for the multivariable H-functionrdquo VijnanaParishad Anusandhan Patrika vol 49 no 2 pp 159ndash175 2006

[10] D L Suthar and P Agarwal ldquoGeneralized mittag -leffler func-tion and the multivariable h-function involving the generalizedmellin-barnes contour integralsrdquoCommunications inNumericalAnalysis vol 2017 no 1 pp 25ndash33 2017

[11] D L Suthar B Debalkie and M Andualem ldquoModifiedSaigo fractional integral operators involving multivariableH-function and general class of multivariable polynomialsrdquoAdvances in Difference Equations vol 2019 no 1 article no 2132019

[12] D L Suthar H Habenom and H Tadesse ldquoCertain integralsinvolving aleph function and Wrightrsquos generalized hypergeo-metric functionrdquo Acta Universitatis Apulensis no 52 pp 1ndash102017

[13] D L Suthar G Reddy and B Shimelis ldquoOn Certain UnifiedFractional Integrals Pertaining to Product of Srivastavarsquos Poly-nomials and N-Functionrdquo International Journal of MathematicsTrends and Technology vol 47 no 1 pp 66ndash73 2017

[14] E D Rainville Special Functions The Macmillan CompanyNew York NY USA 1960

10 Journal of Applied Mathematics

[15] G Szego Orthogonal Polynomials American MathematicalSociety Colloquium Publications vol 23 American Mathemati-cal Society Providence RI USA 1959

[16] A Erdersquolyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol I McGraw-Hill NewYork-Toronto-London 1953

[17] V S Kiryakova Generalized Fractional Calculus and Applica-tions vol 301 of Pitman Research Notes in Mathematics UnitedStates with JohnWiley amp Sons Longman Scientific ampTechnicalNew York NY USA 1994

[18] V P Saxena ldquoFormal solution of certain new pair of dualintegral equations involving H-functionsrdquo Proceedings of theNational Academy of Sciences vol A52 no 3 pp 366ndash375 1982

[19] H M Srivastava ldquoA multilinear generating function for theKonhauser sets of biorthogonal polynomials suggested by theLaguerre polynomialsrdquo Pacific Journal of Mathematics vol 117no 1 pp 183ndash191 1985

[20] H M Srivastava and N P Singh ldquoThe integration of certainproducts of the multivariableH-function with a general class ofpolynomialsrdquo Rendiconti del Circolo Matematico di Palermo vol 32 no 2 pp 157ndash187 1983

[21] F Ayant ldquoThe integration of certains products of special func-tions and multivariable Aleph-functionsrdquo International Journalof Mathematics Trends and Technology vol 31 no 2 pp 101ndash1122016

[22] D Kumar R K Saxena and J Ram ldquoFinite integral formulasinvolving aleph functionrdquo Boletim da Sociedade Paranaense deMatematica vol 36 no 1 pp 177ndash193 2018

[23] D L Suthar S Agarwal and D Kumar ldquoCertain integralsinvolving the product of Gaussian hypergeometric function andAleph functionrdquoHonamMathematical Journal vol 41 no 1 pp1ndash17 2019

[24] D L Suthar and B Debalkie ldquoA new class of integral relationinvolving Aleph-functionsrdquo Surveys in Mathematics and itsApplications vol 12 pp 193ndash201 2017

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Submit your manuscripts atwwwhindawicom

6 Journal of Applied Mathematics

1198688 = 2120588+120590+1 Γ (1 + 120583 + 119899)119899

infinsum119896=0

(minus119899)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120583 + 119896) 119896times 1(2120587120596)119903 int1198711 sdot sdot sdotsdot int

119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (2119897119894+ℎ119894119911119894)120585119894

times Γ (1 + 120588 + 119896 + sum119903119894=1 119897119894120585119894) Γ (1 + 120590 + sum119903

119894=1 ℎ119894120585119894)Γ (2 + 120588 + 120590 + 119896 + sum119903119894=1 (119897119894 + ℎ119894) 120585119894) 1198891205851

119889120585119903(34)

Finally rewriting the above equation by virtue of (1) we arriveat

1198688 = 2120588+120590+1 Γ (1 + 120583 + 119899)119899

infinsum119896=0

(minus119899)119896 (1 + 120583 + V + 119899)119896Γ (1 + 120583 + 119896) 119896

times alefsym0119899+2119880119901119894+2119902119894+1120591119894 119877119881119882

[[[[[

21198971+ℎ111991112119897119903+ℎ119903119911119903

| (minus120588 minus 119896 1198971 119897119903) (minus120590 ℎ1 ℎ119903) 119860 119861(minus1 minus 120588 minus 120590 minus 119896) (1198971 + ℎ1) (119897119903 + ℎ119903)) 119862 119863

]]]]]

(35)

ThusR(V) gt minus1R(120583) gt minus1 and |arg 119911| lt (12)120587Ω4 Integrals Involving Multivariable Aleph-Function with Legendre Function

The solution of Legendre differential equation in the form ofGauss hypergeometric type is as follows

119901120583V (119909) = 1Γ (1 minus 120583) (119909 + 1119909 minus 1)

12120583

sdot 21198651 [minusV V + 1 1 minus 120583 1 minus 1199092 ] |1 minus 119909| lt 2

(36)

Here119901120583V (119909) is known as the Legendre function of the first kind

[16]

Next we derive the integrals with Legendre function

1198689 = int1

0119909120590minus1 (1 minus 1199092)1205832 119901120583

V (119909)sdot alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205881 119911119903119909120588119903] 119889119909 = 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

0119909120590+sum119903119894=1 120588119894120585119894minus1 (1 minus 1199092)1205832 119875120583

V (119909) 1198891199091198891205851 119889120585119903

(37)

Next we use the following formula ([16] Sec 312) for 120583 isinNR(120590) gt 0

int1

0119909120590minus1 (1 minus 1199092)1205832 119901120583

V (119909) 119889119909 = (minus1)120583 2minus120590minus12058312058712Γ (120590) Γ (1 + 120583 + V)Γ (1 minus 120583 + V) Γ (12 + 1205902 + 1205832 minus V2) Γ (1 + 1205902 + 1205832 + V2) (38)

Now applying (38) we obtain

1198689 = 2minus120590minus120583 (minus1)120583 12058712 Γ (1 + 120583 + V)Γ (1 minus 120583 + V) 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585119894) (2minus120588119894119911119894)120585119894

Journal of Applied Mathematics 7

times Γ (120590 + sum119903119894=1 120588119894120585119894)Γ (12 + 1205902 + 1205832 minus V2 + (sum119903

119894=1 120588119894120585119894) 2) Γ (1 + 1205902 + 1205832 + V2 + (sum119903119894=1 120588119894120585119894) 2)1198891205851 119889120585119903

(39)

1198689 = 2minus120590minus120583 (minus1)120583 12058712 Γ (1 + 120583 + V)Γ (1 minus 120583 + V)

sdot alefsym0119899+1119880119901119894+1119902119894+2120591119894 119877119881119882

[[[[[

2minus120588111991112minus120588119903119911119903

| (1 minus 120590 1205881 120588119903) 119860 119861(12 minus 1205832 + V2 minus 1205902 12058812 1205881199032 ) (minus1205832 minus V2 minus 1205902 12058812 1205881199032 ) 119862 119863

]]]]]

(40)

Thus |arg (119911)| lt (12)120587Ω 120590 gt 0 and 120583 isin N cup 011986810 = int1

0119909120590minus1 (1 minus 1199092)minus1205832 119901120583

V (119909)sdot alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205881 119911119903119909120588119903] 119889119909 = 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

0119909120590+sum119903119894=1 120588119894120585119894minus1 (1 minus 1199092)minus1205832 119875120583

V (119909) 1198891199091198891205851

119889120585119903(41)

Next we use the following formula ([16] Sec 312) for 120583 isinNR(120590) gt 0

int1

0119909120590minus1 (1 minus 1199092)minus1205832 119901120583

V (119909) 119889119909= 2minus120590+12058312058712Γ (120590)Γ (12 + 1205902 minus 1205832 minus V2) Γ (1 + 1205902 minus 1205832 minus V2)

(42)

Using (42) we obtain

11986810 = 2minus120590+12058312058712 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (2minus120588119894119911119894)120585119894

times Γ (120590 + sum119903119894=1 120588119894120585119894)Γ (12 + 1205902 minus 1205832 minus V2 + (sum119903

119894=1 120588119894120585119894) 2) Γ (1 + 1205902 minus 1205832 minus V2 + (sum119903119894=1 120588119894120585119894) 2)1198891205851 119889120585119903

(43)

11986810 = 2minus120590+12058312058712alefsym0119899+1119880119901119894+1119902119894+2120591119894119877119881119882

[[[[[

2minus120588111991112minus120588119903119911119903

| (1 minus 120590 1205881 120588119903) 119860 119861(12 + 1205832 + V2 minus 1205902 12058812 1205881199032 ) (1205832 + V2 minus 1205902 12058812 1205881199032 ) 119862 119863

]]]]] (44)

Thus |arg (119911)| lt (12)120587ΩR(120590) gt 0 andR(120583) isin N cup 0

5 Integrals Involving Multivariable Aleph-Function and Bessel-Maitland Function

The Bessel-Maitland function is defined by Kiryakova [17] asfollows

119869120583V (119909) = 120601 (120583 V + 1 119909) = infinsum119896=0

1Γ (120583119896 + V + 1) (minus119909)119896

119896 120583 gt minus1 119909 isin C

(45)

11986811 = intinfin

0119909120588119869120583V (119909)

sdot alefsym0119899119880119901119894119902119894120591119894 119877119881119882

[11991111199091205901 119911119903119909120590119903] 119889119909 119889119909= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585i) 119911120585119894119894

sdot intinfin

0119909120588+sum119903119894=1 120590119894120585119894119869120583V (119909) 119889119909 1198891205851 119889120585119903

(46)

Now using the following formula from [18]

8 Journal of Applied Mathematics

intinfin

0119909120588119869120583V (119909) 119889119909 = Γ (120588 + 1)

Γ (1 + V minus 120583 minus 120583120588)(R (120588) gt minus1 0 lt 120583 lt 1)

(47)and further applying (47) we obtain

11986814 = 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 Γ (1 + 120588 + sum119903119894=1 120590119894120585119894)Γ (1 + V minus 120583 minus 120583120588 minus 120583sum119903

119894=1 120590119894120585119894)1198891205851 119889120585119903 (48)

11986814 = alefsym0119899+1119880119901119894+2119902119894 120591119894119877119881119882

[[[[[

1199111119911119903

| (minus120588 1205901 120590119903) (1 + V minus 120583 minus 120583120588 1205831205901 120583120590119903) 119860 119861119862 119863

]]]]] (49)

Thus |arg 119911| lt (12)120587Ω 120590 minus 120583120590 gt 0 120590 gt 0 0 lt 120583 lt 1 andR(120588 + 1) gt 06 Integrals Involving Multivariable Aleph-Function and General Class of Polynomials

The general class of polynomials 1198781198981 1198981199031198991 119899119903(119909) defined by

Srivastava [19] is as follows

1198781198981 1198981199031198991 119899119903(119909) = [11989911198981]sum

119897119894=0

sdot sdot sdot [119899119903119898119903]sum119897119903=0

119903prod119894=1

(minus119899119894)119898119894119897119894119897119894 119860119899119894 119897119894119909119897119894 (50)

Here 1198991 1198992 119899119903 isin N0 1198981 1198982 119898119903 isin N and thecoefficients 119860119899119894 119897119894

(119899119894 119897119894 ge 0) are any constant numbers Bysuitable restriction of the coefficient 119860119899119894 119897119894

the general classof polynomials has various special cases These include theJacobi polynomials the Laguerre polynomials the Besselpolynomials the Hermite polynomials the Gould-Hopperpolynomials and the Brafman polynomials ([20] p 158-161)

Next we establish the following integral

11986812 = int1

minus1(1 minus 119909)120588minus1 (1 + 119909)120590minus1 1198781198981 11989811990310158401198991 1198991199031015840

[119910 (1 minus 119909)120583 (1+ 119909)V] times alefsym0119899119880

119901119894119902119894 120591119894119877119881119882(1199111 (1 minus 119909)ℎ1

sdot (1 + 119909)1198961 119911119903 (1 minus 119909)ℎ119903 (1 + 119909)119896119903) 119889119909= [11989911198981]sum

1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895 119897119895119910119897119895 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 sdot sdot sdot 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

minus1(1 minus 119909)120588+sum119903119894=1 ℎ119894120585119894+120583sum1199031015840119895=1 119897119895minus1

sdot (1 + 119909)120590+sum119903119894=1 119896119894120585119894+Vsum1199031015840119895=1 119897119895minus1 1198891199091198891205851 119889120585119903

(51)

By applying (21) we have

11986812 = 2120588+120590minus1[11989911198981]sum1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895119897119895(2120583+V119910)119897119895 times 1(2120587120596)119903 int1198711 sdot sdot sdot

sdot int119871119903

120595 (1205851 sdot sdot sdot 120585119903) 119903prod119894=1

120601119894 (120585119894) (2ℎ119894+119896119894119911119894)120585119894 times Γ (120588 + sum119903119894=1 ℎ119894120585119894 + 120583sum1199031015840

119895=1 119897119895) Γ (120590 + sum119903119894=1 119896119894120585119894 + Vsum1199031015840

119895=1 119897119895)Γ (120588 + 120590 + sum119903

119894=1 (ℎ119894 + 119896119894) 120585119894 + (120583 + V)sum1199031015840

119895=1 119897119895) 1198891205851 119889120585119903(52)

11986812 = 2120588+120590minus1[11989911198981]sum1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895119897119895(2120583+V119910)119897119895

sdot alefsym0119899+1119880119901119894+2119902119894 120591119894119877119881119882

[[[[[[

2ℎ1+119896111991112ℎ119903+119896119903119911119903

| (1 minus 120588 minus 120583 1199031015840sum119895=1

119897119895 ℎ1 ℎ119903) (1 minus 120590 minus V1199031015840sum119895=1

119897119895 1198961 119896119903) 119860 119861(1 minus 120588 minus 120590 minus (120583 + V) 1199031015840sum

119895=1119897119895 (ℎ1 + 1198961) (ℎ119903 + 119896119903)) 119862 119863

]]]]]]

(53)

Journal of Applied Mathematics 9

These converge under the following conditions

(1) |arg 119911| lt (12)120587Ω(2) 120588 ge 1 120590 ge 1 120583 ge 0 V ge 0 ℎ ge 0 119896 ge 0 (ℎ and 119896 are

not both zero simultaneously)(3) R(120588) + ℎmin[R(119887119895119861119895)] gt 0 and R(120590) +119896min[R(119887119895119861119895)] gt 0

7 Special Cases

(i)By setting 120575 by 120578minus1 and120588 = 120590 = 120583 = V = 0 (32) transformsto the following integral

11986813 = 2120578 infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 119898)119896 (1 + 119899)119896Γ (1 + 119896) Γ (1 + 119896) (119896)2

times alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

211989711991112119897119911119903

| (1 minus 120578 minus 2119896 1198971 119897119903) 119860 119861(minus120578 minus 2119896 1198971 119897119903) 119862 119863

]]]]](54)

Thus |arg 119911| lt (12)120587Ω(ii)When we substitute 120588 by 120588 minus 1 and 120590 by 120590 minus 1 and put120583 = V = 0 integral (35) transforms to the following

11986814 = 2120588+120590minus1 infinsum119896=0

(minus119899)119896 (1 + 119899)119896Γ (1 + 119896) 119896 times alefsym0119899+2119880119901119894+2119902119894+1120591119894 119877119881119882

[[[[[

21198971+ℎ111991112119897119903+ℎ119903119911119903

| (1 minus 120588 minus 119896 1198971 119897119903) (1 minus 120590 ℎ1 ℎ119903) 119860 119861(1 minus 120588 minus 120590 minus 119896 (1198971 + ℎ1) (119897119903 + ℎ119903)) 119862 119863

]]]]] (55)

Thus |arg 119911| lt (12)120587Ω8 Concluding Remarks

In the present paper we evaluated new integrals involving themultivariable Aleph-function with certain special functionsCertain special cases of integrals that follow Remarks 1ndash3have been investigated in the literature by a number ofauthors [21ndash24]) with different arguments Therefore theresults presented in this paper are easily converted in termsof a similar type of new interesting integrals with differentarguments after some suitable parameter substitutions Inthis sequel one can obtain integral representation of moregeneralized special function which has many applications inphysics and engineering science

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C K Sharma and S S Ahmad ldquoOn the multivariable I-functionrdquo Acta Ciencia Indica Mathematics vol 20 no 2 pp113ndash116 1994

[2] Y N Prasad ldquoMultivariable I-functionrdquo Vijnana ParishadAnusandhan Patrika vol 29 pp 231ndash235 1986

[3] C K Sharma and G K Mishra ldquoExponential Fourier series forthe multivariable I-functionrdquo Acta Ciencia Indica Mathematicsvol 21 no 4 pp 495ndash501 1995

[4] H M Srivastava and R Panda ldquoSome expansion theoremsand generating relations for the H function of several complex

variablesrdquo Commentarii Mathematici Universitatis Sancti Paulivol 24 no 2 pp 119ndash137 1975

[5] N Sudland B Baumann andT FNonnenmacher ldquoOpen prob-lem Who knows about the Aleph () -functions rdquo FractionalCalculus and Applied Analysis vol 1 no 4 pp 401-402 1998

[6] D Kumar S D Purohit and J Choi ldquoGeneralized fractionalintegrals involving product of multivariable H-function and ageneral class of polynomialsrdquo Journal of Nonlinear Sciences andApplications vol 09 no 01 pp 8ndash21 2016

[7] R K Saxena and T K Pogany ldquoOn fractional integrationformulae for Aleph functionsrdquo Applied Mathematics and Com-putation vol 218 no 3 pp 985ndash990 2011

[8] R K Saxena J Ram and D Kumar ldquoGeneralized fractionalintegral of the product of two Aleph-functionsrdquo Applicationsand Applied Mathematics An International Journal (AAM) vol8 no 2 pp 631ndash646 2013

[9] R K Saxena J Ram and D L Suthar ldquoUnified fractionalderivative formulas for the multivariable H-functionrdquo VijnanaParishad Anusandhan Patrika vol 49 no 2 pp 159ndash175 2006

[10] D L Suthar and P Agarwal ldquoGeneralized mittag -leffler func-tion and the multivariable h-function involving the generalizedmellin-barnes contour integralsrdquoCommunications inNumericalAnalysis vol 2017 no 1 pp 25ndash33 2017

[11] D L Suthar B Debalkie and M Andualem ldquoModifiedSaigo fractional integral operators involving multivariableH-function and general class of multivariable polynomialsrdquoAdvances in Difference Equations vol 2019 no 1 article no 2132019

[12] D L Suthar H Habenom and H Tadesse ldquoCertain integralsinvolving aleph function and Wrightrsquos generalized hypergeo-metric functionrdquo Acta Universitatis Apulensis no 52 pp 1ndash102017

[13] D L Suthar G Reddy and B Shimelis ldquoOn Certain UnifiedFractional Integrals Pertaining to Product of Srivastavarsquos Poly-nomials and N-Functionrdquo International Journal of MathematicsTrends and Technology vol 47 no 1 pp 66ndash73 2017

[14] E D Rainville Special Functions The Macmillan CompanyNew York NY USA 1960

10 Journal of Applied Mathematics

[15] G Szego Orthogonal Polynomials American MathematicalSociety Colloquium Publications vol 23 American Mathemati-cal Society Providence RI USA 1959

[16] A Erdersquolyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol I McGraw-Hill NewYork-Toronto-London 1953

[17] V S Kiryakova Generalized Fractional Calculus and Applica-tions vol 301 of Pitman Research Notes in Mathematics UnitedStates with JohnWiley amp Sons Longman Scientific ampTechnicalNew York NY USA 1994

[18] V P Saxena ldquoFormal solution of certain new pair of dualintegral equations involving H-functionsrdquo Proceedings of theNational Academy of Sciences vol A52 no 3 pp 366ndash375 1982

[19] H M Srivastava ldquoA multilinear generating function for theKonhauser sets of biorthogonal polynomials suggested by theLaguerre polynomialsrdquo Pacific Journal of Mathematics vol 117no 1 pp 183ndash191 1985

[20] H M Srivastava and N P Singh ldquoThe integration of certainproducts of the multivariableH-function with a general class ofpolynomialsrdquo Rendiconti del Circolo Matematico di Palermo vol 32 no 2 pp 157ndash187 1983

[21] F Ayant ldquoThe integration of certains products of special func-tions and multivariable Aleph-functionsrdquo International Journalof Mathematics Trends and Technology vol 31 no 2 pp 101ndash1122016

[22] D Kumar R K Saxena and J Ram ldquoFinite integral formulasinvolving aleph functionrdquo Boletim da Sociedade Paranaense deMatematica vol 36 no 1 pp 177ndash193 2018

[23] D L Suthar S Agarwal and D Kumar ldquoCertain integralsinvolving the product of Gaussian hypergeometric function andAleph functionrdquoHonamMathematical Journal vol 41 no 1 pp1ndash17 2019

[24] D L Suthar and B Debalkie ldquoA new class of integral relationinvolving Aleph-functionsrdquo Surveys in Mathematics and itsApplications vol 12 pp 193ndash201 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Journal of Applied Mathematics 7

times Γ (120590 + sum119903119894=1 120588119894120585119894)Γ (12 + 1205902 + 1205832 minus V2 + (sum119903

119894=1 120588119894120585119894) 2) Γ (1 + 1205902 + 1205832 + V2 + (sum119903119894=1 120588119894120585119894) 2)1198891205851 119889120585119903

(39)

1198689 = 2minus120590minus120583 (minus1)120583 12058712 Γ (1 + 120583 + V)Γ (1 minus 120583 + V)

sdot alefsym0119899+1119880119901119894+1119902119894+2120591119894 119877119881119882

[[[[[

2minus120588111991112minus120588119903119911119903

| (1 minus 120590 1205881 120588119903) 119860 119861(12 minus 1205832 + V2 minus 1205902 12058812 1205881199032 ) (minus1205832 minus V2 minus 1205902 12058812 1205881199032 ) 119862 119863

]]]]]

(40)

Thus |arg (119911)| lt (12)120587Ω 120590 gt 0 and 120583 isin N cup 011986810 = int1

0119909120590minus1 (1 minus 1199092)minus1205832 119901120583

V (119909)sdot alefsym0119899119880

119901119894119902119894 120591119894119877119881119882[11991111199091205881 119911119903119909120588119903] 119889119909 = 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

0119909120590+sum119903119894=1 120588119894120585119894minus1 (1 minus 1199092)minus1205832 119875120583

V (119909) 1198891199091198891205851

119889120585119903(41)

Next we use the following formula ([16] Sec 312) for 120583 isinNR(120590) gt 0

int1

0119909120590minus1 (1 minus 1199092)minus1205832 119901120583

V (119909) 119889119909= 2minus120590+12058312058712Γ (120590)Γ (12 + 1205902 minus 1205832 minus V2) Γ (1 + 1205902 minus 1205832 minus V2)

(42)

Using (42) we obtain

11986810 = 2minus120590+12058312058712 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) (2minus120588119894119911119894)120585119894

times Γ (120590 + sum119903119894=1 120588119894120585119894)Γ (12 + 1205902 minus 1205832 minus V2 + (sum119903

119894=1 120588119894120585119894) 2) Γ (1 + 1205902 minus 1205832 minus V2 + (sum119903119894=1 120588119894120585119894) 2)1198891205851 119889120585119903

(43)

11986810 = 2minus120590+12058312058712alefsym0119899+1119880119901119894+1119902119894+2120591119894119877119881119882

[[[[[

2minus120588111991112minus120588119903119911119903

| (1 minus 120590 1205881 120588119903) 119860 119861(12 + 1205832 + V2 minus 1205902 12058812 1205881199032 ) (1205832 + V2 minus 1205902 12058812 1205881199032 ) 119862 119863

]]]]] (44)

Thus |arg (119911)| lt (12)120587ΩR(120590) gt 0 andR(120583) isin N cup 0

5 Integrals Involving Multivariable Aleph-Function and Bessel-Maitland Function

The Bessel-Maitland function is defined by Kiryakova [17] asfollows

119869120583V (119909) = 120601 (120583 V + 1 119909) = infinsum119896=0

1Γ (120583119896 + V + 1) (minus119909)119896

119896 120583 gt minus1 119909 isin C

(45)

11986811 = intinfin

0119909120588119869120583V (119909)

sdot alefsym0119899119880119901119894119902119894120591119894 119877119881119882

[11991111199091205901 119911119903119909120590119903] 119889119909 119889119909= 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod

119894=1

120601119894 (120585i) 119911120585119894119894

sdot intinfin

0119909120588+sum119903119894=1 120590119894120585119894119869120583V (119909) 119889119909 1198891205851 119889120585119903

(46)

Now using the following formula from [18]

8 Journal of Applied Mathematics

intinfin

0119909120588119869120583V (119909) 119889119909 = Γ (120588 + 1)

Γ (1 + V minus 120583 minus 120583120588)(R (120588) gt minus1 0 lt 120583 lt 1)

(47)and further applying (47) we obtain

11986814 = 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 Γ (1 + 120588 + sum119903119894=1 120590119894120585119894)Γ (1 + V minus 120583 minus 120583120588 minus 120583sum119903

119894=1 120590119894120585119894)1198891205851 119889120585119903 (48)

11986814 = alefsym0119899+1119880119901119894+2119902119894 120591119894119877119881119882

[[[[[

1199111119911119903

| (minus120588 1205901 120590119903) (1 + V minus 120583 minus 120583120588 1205831205901 120583120590119903) 119860 119861119862 119863

]]]]] (49)

Thus |arg 119911| lt (12)120587Ω 120590 minus 120583120590 gt 0 120590 gt 0 0 lt 120583 lt 1 andR(120588 + 1) gt 06 Integrals Involving Multivariable Aleph-Function and General Class of Polynomials

The general class of polynomials 1198781198981 1198981199031198991 119899119903(119909) defined by

Srivastava [19] is as follows

1198781198981 1198981199031198991 119899119903(119909) = [11989911198981]sum

119897119894=0

sdot sdot sdot [119899119903119898119903]sum119897119903=0

119903prod119894=1

(minus119899119894)119898119894119897119894119897119894 119860119899119894 119897119894119909119897119894 (50)

Here 1198991 1198992 119899119903 isin N0 1198981 1198982 119898119903 isin N and thecoefficients 119860119899119894 119897119894

(119899119894 119897119894 ge 0) are any constant numbers Bysuitable restriction of the coefficient 119860119899119894 119897119894

the general classof polynomials has various special cases These include theJacobi polynomials the Laguerre polynomials the Besselpolynomials the Hermite polynomials the Gould-Hopperpolynomials and the Brafman polynomials ([20] p 158-161)

Next we establish the following integral

11986812 = int1

minus1(1 minus 119909)120588minus1 (1 + 119909)120590minus1 1198781198981 11989811990310158401198991 1198991199031015840

[119910 (1 minus 119909)120583 (1+ 119909)V] times alefsym0119899119880

119901119894119902119894 120591119894119877119881119882(1199111 (1 minus 119909)ℎ1

sdot (1 + 119909)1198961 119911119903 (1 minus 119909)ℎ119903 (1 + 119909)119896119903) 119889119909= [11989911198981]sum

1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895 119897119895119910119897119895 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 sdot sdot sdot 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

minus1(1 minus 119909)120588+sum119903119894=1 ℎ119894120585119894+120583sum1199031015840119895=1 119897119895minus1

sdot (1 + 119909)120590+sum119903119894=1 119896119894120585119894+Vsum1199031015840119895=1 119897119895minus1 1198891199091198891205851 119889120585119903

(51)

By applying (21) we have

11986812 = 2120588+120590minus1[11989911198981]sum1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895119897119895(2120583+V119910)119897119895 times 1(2120587120596)119903 int1198711 sdot sdot sdot

sdot int119871119903

120595 (1205851 sdot sdot sdot 120585119903) 119903prod119894=1

120601119894 (120585119894) (2ℎ119894+119896119894119911119894)120585119894 times Γ (120588 + sum119903119894=1 ℎ119894120585119894 + 120583sum1199031015840

119895=1 119897119895) Γ (120590 + sum119903119894=1 119896119894120585119894 + Vsum1199031015840

119895=1 119897119895)Γ (120588 + 120590 + sum119903

119894=1 (ℎ119894 + 119896119894) 120585119894 + (120583 + V)sum1199031015840

119895=1 119897119895) 1198891205851 119889120585119903(52)

11986812 = 2120588+120590minus1[11989911198981]sum1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895119897119895(2120583+V119910)119897119895

sdot alefsym0119899+1119880119901119894+2119902119894 120591119894119877119881119882

[[[[[[

2ℎ1+119896111991112ℎ119903+119896119903119911119903

| (1 minus 120588 minus 120583 1199031015840sum119895=1

119897119895 ℎ1 ℎ119903) (1 minus 120590 minus V1199031015840sum119895=1

119897119895 1198961 119896119903) 119860 119861(1 minus 120588 minus 120590 minus (120583 + V) 1199031015840sum

119895=1119897119895 (ℎ1 + 1198961) (ℎ119903 + 119896119903)) 119862 119863

]]]]]]

(53)

Journal of Applied Mathematics 9

These converge under the following conditions

(1) |arg 119911| lt (12)120587Ω(2) 120588 ge 1 120590 ge 1 120583 ge 0 V ge 0 ℎ ge 0 119896 ge 0 (ℎ and 119896 are

not both zero simultaneously)(3) R(120588) + ℎmin[R(119887119895119861119895)] gt 0 and R(120590) +119896min[R(119887119895119861119895)] gt 0

7 Special Cases

(i)By setting 120575 by 120578minus1 and120588 = 120590 = 120583 = V = 0 (32) transformsto the following integral

11986813 = 2120578 infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 119898)119896 (1 + 119899)119896Γ (1 + 119896) Γ (1 + 119896) (119896)2

times alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

211989711991112119897119911119903

| (1 minus 120578 minus 2119896 1198971 119897119903) 119860 119861(minus120578 minus 2119896 1198971 119897119903) 119862 119863

]]]]](54)

Thus |arg 119911| lt (12)120587Ω(ii)When we substitute 120588 by 120588 minus 1 and 120590 by 120590 minus 1 and put120583 = V = 0 integral (35) transforms to the following

11986814 = 2120588+120590minus1 infinsum119896=0

(minus119899)119896 (1 + 119899)119896Γ (1 + 119896) 119896 times alefsym0119899+2119880119901119894+2119902119894+1120591119894 119877119881119882

[[[[[

21198971+ℎ111991112119897119903+ℎ119903119911119903

| (1 minus 120588 minus 119896 1198971 119897119903) (1 minus 120590 ℎ1 ℎ119903) 119860 119861(1 minus 120588 minus 120590 minus 119896 (1198971 + ℎ1) (119897119903 + ℎ119903)) 119862 119863

]]]]] (55)

Thus |arg 119911| lt (12)120587Ω8 Concluding Remarks

In the present paper we evaluated new integrals involving themultivariable Aleph-function with certain special functionsCertain special cases of integrals that follow Remarks 1ndash3have been investigated in the literature by a number ofauthors [21ndash24]) with different arguments Therefore theresults presented in this paper are easily converted in termsof a similar type of new interesting integrals with differentarguments after some suitable parameter substitutions Inthis sequel one can obtain integral representation of moregeneralized special function which has many applications inphysics and engineering science

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C K Sharma and S S Ahmad ldquoOn the multivariable I-functionrdquo Acta Ciencia Indica Mathematics vol 20 no 2 pp113ndash116 1994

[2] Y N Prasad ldquoMultivariable I-functionrdquo Vijnana ParishadAnusandhan Patrika vol 29 pp 231ndash235 1986

[3] C K Sharma and G K Mishra ldquoExponential Fourier series forthe multivariable I-functionrdquo Acta Ciencia Indica Mathematicsvol 21 no 4 pp 495ndash501 1995

[4] H M Srivastava and R Panda ldquoSome expansion theoremsand generating relations for the H function of several complex

variablesrdquo Commentarii Mathematici Universitatis Sancti Paulivol 24 no 2 pp 119ndash137 1975

[5] N Sudland B Baumann andT FNonnenmacher ldquoOpen prob-lem Who knows about the Aleph () -functions rdquo FractionalCalculus and Applied Analysis vol 1 no 4 pp 401-402 1998

[6] D Kumar S D Purohit and J Choi ldquoGeneralized fractionalintegrals involving product of multivariable H-function and ageneral class of polynomialsrdquo Journal of Nonlinear Sciences andApplications vol 09 no 01 pp 8ndash21 2016

[7] R K Saxena and T K Pogany ldquoOn fractional integrationformulae for Aleph functionsrdquo Applied Mathematics and Com-putation vol 218 no 3 pp 985ndash990 2011

[8] R K Saxena J Ram and D Kumar ldquoGeneralized fractionalintegral of the product of two Aleph-functionsrdquo Applicationsand Applied Mathematics An International Journal (AAM) vol8 no 2 pp 631ndash646 2013

[9] R K Saxena J Ram and D L Suthar ldquoUnified fractionalderivative formulas for the multivariable H-functionrdquo VijnanaParishad Anusandhan Patrika vol 49 no 2 pp 159ndash175 2006

[10] D L Suthar and P Agarwal ldquoGeneralized mittag -leffler func-tion and the multivariable h-function involving the generalizedmellin-barnes contour integralsrdquoCommunications inNumericalAnalysis vol 2017 no 1 pp 25ndash33 2017

[11] D L Suthar B Debalkie and M Andualem ldquoModifiedSaigo fractional integral operators involving multivariableH-function and general class of multivariable polynomialsrdquoAdvances in Difference Equations vol 2019 no 1 article no 2132019

[12] D L Suthar H Habenom and H Tadesse ldquoCertain integralsinvolving aleph function and Wrightrsquos generalized hypergeo-metric functionrdquo Acta Universitatis Apulensis no 52 pp 1ndash102017

[13] D L Suthar G Reddy and B Shimelis ldquoOn Certain UnifiedFractional Integrals Pertaining to Product of Srivastavarsquos Poly-nomials and N-Functionrdquo International Journal of MathematicsTrends and Technology vol 47 no 1 pp 66ndash73 2017

[14] E D Rainville Special Functions The Macmillan CompanyNew York NY USA 1960

10 Journal of Applied Mathematics

[15] G Szego Orthogonal Polynomials American MathematicalSociety Colloquium Publications vol 23 American Mathemati-cal Society Providence RI USA 1959

[16] A Erdersquolyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol I McGraw-Hill NewYork-Toronto-London 1953

[17] V S Kiryakova Generalized Fractional Calculus and Applica-tions vol 301 of Pitman Research Notes in Mathematics UnitedStates with JohnWiley amp Sons Longman Scientific ampTechnicalNew York NY USA 1994

[18] V P Saxena ldquoFormal solution of certain new pair of dualintegral equations involving H-functionsrdquo Proceedings of theNational Academy of Sciences vol A52 no 3 pp 366ndash375 1982

[19] H M Srivastava ldquoA multilinear generating function for theKonhauser sets of biorthogonal polynomials suggested by theLaguerre polynomialsrdquo Pacific Journal of Mathematics vol 117no 1 pp 183ndash191 1985

[20] H M Srivastava and N P Singh ldquoThe integration of certainproducts of the multivariableH-function with a general class ofpolynomialsrdquo Rendiconti del Circolo Matematico di Palermo vol 32 no 2 pp 157ndash187 1983

[21] F Ayant ldquoThe integration of certains products of special func-tions and multivariable Aleph-functionsrdquo International Journalof Mathematics Trends and Technology vol 31 no 2 pp 101ndash1122016

[22] D Kumar R K Saxena and J Ram ldquoFinite integral formulasinvolving aleph functionrdquo Boletim da Sociedade Paranaense deMatematica vol 36 no 1 pp 177ndash193 2018

[23] D L Suthar S Agarwal and D Kumar ldquoCertain integralsinvolving the product of Gaussian hypergeometric function andAleph functionrdquoHonamMathematical Journal vol 41 no 1 pp1ndash17 2019

[24] D L Suthar and B Debalkie ldquoA new class of integral relationinvolving Aleph-functionsrdquo Surveys in Mathematics and itsApplications vol 12 pp 193ndash201 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Journal of Applied Mathematics

intinfin

0119909120588119869120583V (119909) 119889119909 = Γ (120588 + 1)

Γ (1 + V minus 120583 minus 120583120588)(R (120588) gt minus1 0 lt 120583 lt 1)

(47)and further applying (47) we obtain

11986814 = 1(2120587120596)119903 int1198711 sdot sdot sdot int119871119903 120595 (1205851 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894 Γ (1 + 120588 + sum119903119894=1 120590119894120585119894)Γ (1 + V minus 120583 minus 120583120588 minus 120583sum119903

119894=1 120590119894120585119894)1198891205851 119889120585119903 (48)

11986814 = alefsym0119899+1119880119901119894+2119902119894 120591119894119877119881119882

[[[[[

1199111119911119903

| (minus120588 1205901 120590119903) (1 + V minus 120583 minus 120583120588 1205831205901 120583120590119903) 119860 119861119862 119863

]]]]] (49)

Thus |arg 119911| lt (12)120587Ω 120590 minus 120583120590 gt 0 120590 gt 0 0 lt 120583 lt 1 andR(120588 + 1) gt 06 Integrals Involving Multivariable Aleph-Function and General Class of Polynomials

The general class of polynomials 1198781198981 1198981199031198991 119899119903(119909) defined by

Srivastava [19] is as follows

1198781198981 1198981199031198991 119899119903(119909) = [11989911198981]sum

119897119894=0

sdot sdot sdot [119899119903119898119903]sum119897119903=0

119903prod119894=1

(minus119899119894)119898119894119897119894119897119894 119860119899119894 119897119894119909119897119894 (50)

Here 1198991 1198992 119899119903 isin N0 1198981 1198982 119898119903 isin N and thecoefficients 119860119899119894 119897119894

(119899119894 119897119894 ge 0) are any constant numbers Bysuitable restriction of the coefficient 119860119899119894 119897119894

the general classof polynomials has various special cases These include theJacobi polynomials the Laguerre polynomials the Besselpolynomials the Hermite polynomials the Gould-Hopperpolynomials and the Brafman polynomials ([20] p 158-161)

Next we establish the following integral

11986812 = int1

minus1(1 minus 119909)120588minus1 (1 + 119909)120590minus1 1198781198981 11989811990310158401198991 1198991199031015840

[119910 (1 minus 119909)120583 (1+ 119909)V] times alefsym0119899119880

119901119894119902119894 120591119894119877119881119882(1199111 (1 minus 119909)ℎ1

sdot (1 + 119909)1198961 119911119903 (1 minus 119909)ℎ119903 (1 + 119909)119896119903) 119889119909= [11989911198981]sum

1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895 119897119895119910119897119895 1(2120587120596)119903

sdot int1198711

sdot sdot sdot int119871119903

120595 (1205851 sdot sdot sdot 120585119903) 119903prod119894=1

120601119894 (120585119894) 119911120585119894119894times int1

minus1(1 minus 119909)120588+sum119903119894=1 ℎ119894120585119894+120583sum1199031015840119895=1 119897119895minus1

sdot (1 + 119909)120590+sum119903119894=1 119896119894120585119894+Vsum1199031015840119895=1 119897119895minus1 1198891199091198891205851 119889120585119903

(51)

By applying (21) we have

11986812 = 2120588+120590minus1[11989911198981]sum1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895119897119895(2120583+V119910)119897119895 times 1(2120587120596)119903 int1198711 sdot sdot sdot

sdot int119871119903

120595 (1205851 sdot sdot sdot 120585119903) 119903prod119894=1

120601119894 (120585119894) (2ℎ119894+119896119894119911119894)120585119894 times Γ (120588 + sum119903119894=1 ℎ119894120585119894 + 120583sum1199031015840

119895=1 119897119895) Γ (120590 + sum119903119894=1 119896119894120585119894 + Vsum1199031015840

119895=1 119897119895)Γ (120588 + 120590 + sum119903

119894=1 (ℎ119894 + 119896119894) 120585119894 + (120583 + V)sum1199031015840

119895=1 119897119895) 1198891205851 119889120585119903(52)

11986812 = 2120588+120590minus1[11989911198981]sum1198971=0

sdot sdot sdot [1198991199031015840 1198981199031015840 ]sum1198971199031015840

1199031015840prod119894=1

(minus119899119895)119898119895119897119895119897119895 119860119899119895119897119895(2120583+V119910)119897119895

sdot alefsym0119899+1119880119901119894+2119902119894 120591119894119877119881119882

[[[[[[

2ℎ1+119896111991112ℎ119903+119896119903119911119903

| (1 minus 120588 minus 120583 1199031015840sum119895=1

119897119895 ℎ1 ℎ119903) (1 minus 120590 minus V1199031015840sum119895=1

119897119895 1198961 119896119903) 119860 119861(1 minus 120588 minus 120590 minus (120583 + V) 1199031015840sum

119895=1119897119895 (ℎ1 + 1198961) (ℎ119903 + 119896119903)) 119862 119863

]]]]]]

(53)

Journal of Applied Mathematics 9

These converge under the following conditions

(1) |arg 119911| lt (12)120587Ω(2) 120588 ge 1 120590 ge 1 120583 ge 0 V ge 0 ℎ ge 0 119896 ge 0 (ℎ and 119896 are

not both zero simultaneously)(3) R(120588) + ℎmin[R(119887119895119861119895)] gt 0 and R(120590) +119896min[R(119887119895119861119895)] gt 0

7 Special Cases

(i)By setting 120575 by 120578minus1 and120588 = 120590 = 120583 = V = 0 (32) transformsto the following integral

11986813 = 2120578 infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 119898)119896 (1 + 119899)119896Γ (1 + 119896) Γ (1 + 119896) (119896)2

times alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

211989711991112119897119911119903

| (1 minus 120578 minus 2119896 1198971 119897119903) 119860 119861(minus120578 minus 2119896 1198971 119897119903) 119862 119863

]]]]](54)

Thus |arg 119911| lt (12)120587Ω(ii)When we substitute 120588 by 120588 minus 1 and 120590 by 120590 minus 1 and put120583 = V = 0 integral (35) transforms to the following

11986814 = 2120588+120590minus1 infinsum119896=0

(minus119899)119896 (1 + 119899)119896Γ (1 + 119896) 119896 times alefsym0119899+2119880119901119894+2119902119894+1120591119894 119877119881119882

[[[[[

21198971+ℎ111991112119897119903+ℎ119903119911119903

| (1 minus 120588 minus 119896 1198971 119897119903) (1 minus 120590 ℎ1 ℎ119903) 119860 119861(1 minus 120588 minus 120590 minus 119896 (1198971 + ℎ1) (119897119903 + ℎ119903)) 119862 119863

]]]]] (55)

Thus |arg 119911| lt (12)120587Ω8 Concluding Remarks

In the present paper we evaluated new integrals involving themultivariable Aleph-function with certain special functionsCertain special cases of integrals that follow Remarks 1ndash3have been investigated in the literature by a number ofauthors [21ndash24]) with different arguments Therefore theresults presented in this paper are easily converted in termsof a similar type of new interesting integrals with differentarguments after some suitable parameter substitutions Inthis sequel one can obtain integral representation of moregeneralized special function which has many applications inphysics and engineering science

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C K Sharma and S S Ahmad ldquoOn the multivariable I-functionrdquo Acta Ciencia Indica Mathematics vol 20 no 2 pp113ndash116 1994

[2] Y N Prasad ldquoMultivariable I-functionrdquo Vijnana ParishadAnusandhan Patrika vol 29 pp 231ndash235 1986

[3] C K Sharma and G K Mishra ldquoExponential Fourier series forthe multivariable I-functionrdquo Acta Ciencia Indica Mathematicsvol 21 no 4 pp 495ndash501 1995

[4] H M Srivastava and R Panda ldquoSome expansion theoremsand generating relations for the H function of several complex

variablesrdquo Commentarii Mathematici Universitatis Sancti Paulivol 24 no 2 pp 119ndash137 1975

[5] N Sudland B Baumann andT FNonnenmacher ldquoOpen prob-lem Who knows about the Aleph () -functions rdquo FractionalCalculus and Applied Analysis vol 1 no 4 pp 401-402 1998

[6] D Kumar S D Purohit and J Choi ldquoGeneralized fractionalintegrals involving product of multivariable H-function and ageneral class of polynomialsrdquo Journal of Nonlinear Sciences andApplications vol 09 no 01 pp 8ndash21 2016

[7] R K Saxena and T K Pogany ldquoOn fractional integrationformulae for Aleph functionsrdquo Applied Mathematics and Com-putation vol 218 no 3 pp 985ndash990 2011

[8] R K Saxena J Ram and D Kumar ldquoGeneralized fractionalintegral of the product of two Aleph-functionsrdquo Applicationsand Applied Mathematics An International Journal (AAM) vol8 no 2 pp 631ndash646 2013

[9] R K Saxena J Ram and D L Suthar ldquoUnified fractionalderivative formulas for the multivariable H-functionrdquo VijnanaParishad Anusandhan Patrika vol 49 no 2 pp 159ndash175 2006

[10] D L Suthar and P Agarwal ldquoGeneralized mittag -leffler func-tion and the multivariable h-function involving the generalizedmellin-barnes contour integralsrdquoCommunications inNumericalAnalysis vol 2017 no 1 pp 25ndash33 2017

[11] D L Suthar B Debalkie and M Andualem ldquoModifiedSaigo fractional integral operators involving multivariableH-function and general class of multivariable polynomialsrdquoAdvances in Difference Equations vol 2019 no 1 article no 2132019

[12] D L Suthar H Habenom and H Tadesse ldquoCertain integralsinvolving aleph function and Wrightrsquos generalized hypergeo-metric functionrdquo Acta Universitatis Apulensis no 52 pp 1ndash102017

[13] D L Suthar G Reddy and B Shimelis ldquoOn Certain UnifiedFractional Integrals Pertaining to Product of Srivastavarsquos Poly-nomials and N-Functionrdquo International Journal of MathematicsTrends and Technology vol 47 no 1 pp 66ndash73 2017

[14] E D Rainville Special Functions The Macmillan CompanyNew York NY USA 1960

10 Journal of Applied Mathematics

[15] G Szego Orthogonal Polynomials American MathematicalSociety Colloquium Publications vol 23 American Mathemati-cal Society Providence RI USA 1959

[16] A Erdersquolyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol I McGraw-Hill NewYork-Toronto-London 1953

[17] V S Kiryakova Generalized Fractional Calculus and Applica-tions vol 301 of Pitman Research Notes in Mathematics UnitedStates with JohnWiley amp Sons Longman Scientific ampTechnicalNew York NY USA 1994

[18] V P Saxena ldquoFormal solution of certain new pair of dualintegral equations involving H-functionsrdquo Proceedings of theNational Academy of Sciences vol A52 no 3 pp 366ndash375 1982

[19] H M Srivastava ldquoA multilinear generating function for theKonhauser sets of biorthogonal polynomials suggested by theLaguerre polynomialsrdquo Pacific Journal of Mathematics vol 117no 1 pp 183ndash191 1985

[20] H M Srivastava and N P Singh ldquoThe integration of certainproducts of the multivariableH-function with a general class ofpolynomialsrdquo Rendiconti del Circolo Matematico di Palermo vol 32 no 2 pp 157ndash187 1983

[21] F Ayant ldquoThe integration of certains products of special func-tions and multivariable Aleph-functionsrdquo International Journalof Mathematics Trends and Technology vol 31 no 2 pp 101ndash1122016

[22] D Kumar R K Saxena and J Ram ldquoFinite integral formulasinvolving aleph functionrdquo Boletim da Sociedade Paranaense deMatematica vol 36 no 1 pp 177ndash193 2018

[23] D L Suthar S Agarwal and D Kumar ldquoCertain integralsinvolving the product of Gaussian hypergeometric function andAleph functionrdquoHonamMathematical Journal vol 41 no 1 pp1ndash17 2019

[24] D L Suthar and B Debalkie ldquoA new class of integral relationinvolving Aleph-functionsrdquo Surveys in Mathematics and itsApplications vol 12 pp 193ndash201 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Journal of Applied Mathematics 9

These converge under the following conditions

(1) |arg 119911| lt (12)120587Ω(2) 120588 ge 1 120590 ge 1 120583 ge 0 V ge 0 ℎ ge 0 119896 ge 0 (ℎ and 119896 are

not both zero simultaneously)(3) R(120588) + ℎmin[R(119887119895119861119895)] gt 0 and R(120590) +119896min[R(119887119895119861119895)] gt 0

7 Special Cases

(i)By setting 120575 by 120578minus1 and120588 = 120590 = 120583 = V = 0 (32) transformsto the following integral

11986813 = 2120578 infinsum119896=0

(minus119898)119896 (minus119899)119896 (1 + 119898)119896 (1 + 119899)119896Γ (1 + 119896) Γ (1 + 119896) (119896)2

times alefsym0119899+1119880119901119894+1119902119894+1120591119894 119877119881119882

[[[[[

211989711991112119897119911119903

| (1 minus 120578 minus 2119896 1198971 119897119903) 119860 119861(minus120578 minus 2119896 1198971 119897119903) 119862 119863

]]]]](54)

Thus |arg 119911| lt (12)120587Ω(ii)When we substitute 120588 by 120588 minus 1 and 120590 by 120590 minus 1 and put120583 = V = 0 integral (35) transforms to the following

11986814 = 2120588+120590minus1 infinsum119896=0

(minus119899)119896 (1 + 119899)119896Γ (1 + 119896) 119896 times alefsym0119899+2119880119901119894+2119902119894+1120591119894 119877119881119882

[[[[[

21198971+ℎ111991112119897119903+ℎ119903119911119903

| (1 minus 120588 minus 119896 1198971 119897119903) (1 minus 120590 ℎ1 ℎ119903) 119860 119861(1 minus 120588 minus 120590 minus 119896 (1198971 + ℎ1) (119897119903 + ℎ119903)) 119862 119863

]]]]] (55)

Thus |arg 119911| lt (12)120587Ω8 Concluding Remarks

In the present paper we evaluated new integrals involving themultivariable Aleph-function with certain special functionsCertain special cases of integrals that follow Remarks 1ndash3have been investigated in the literature by a number ofauthors [21ndash24]) with different arguments Therefore theresults presented in this paper are easily converted in termsof a similar type of new interesting integrals with differentarguments after some suitable parameter substitutions Inthis sequel one can obtain integral representation of moregeneralized special function which has many applications inphysics and engineering science

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C K Sharma and S S Ahmad ldquoOn the multivariable I-functionrdquo Acta Ciencia Indica Mathematics vol 20 no 2 pp113ndash116 1994

[2] Y N Prasad ldquoMultivariable I-functionrdquo Vijnana ParishadAnusandhan Patrika vol 29 pp 231ndash235 1986

[3] C K Sharma and G K Mishra ldquoExponential Fourier series forthe multivariable I-functionrdquo Acta Ciencia Indica Mathematicsvol 21 no 4 pp 495ndash501 1995

[4] H M Srivastava and R Panda ldquoSome expansion theoremsand generating relations for the H function of several complex

variablesrdquo Commentarii Mathematici Universitatis Sancti Paulivol 24 no 2 pp 119ndash137 1975

[5] N Sudland B Baumann andT FNonnenmacher ldquoOpen prob-lem Who knows about the Aleph () -functions rdquo FractionalCalculus and Applied Analysis vol 1 no 4 pp 401-402 1998

[6] D Kumar S D Purohit and J Choi ldquoGeneralized fractionalintegrals involving product of multivariable H-function and ageneral class of polynomialsrdquo Journal of Nonlinear Sciences andApplications vol 09 no 01 pp 8ndash21 2016

[7] R K Saxena and T K Pogany ldquoOn fractional integrationformulae for Aleph functionsrdquo Applied Mathematics and Com-putation vol 218 no 3 pp 985ndash990 2011

[8] R K Saxena J Ram and D Kumar ldquoGeneralized fractionalintegral of the product of two Aleph-functionsrdquo Applicationsand Applied Mathematics An International Journal (AAM) vol8 no 2 pp 631ndash646 2013

[9] R K Saxena J Ram and D L Suthar ldquoUnified fractionalderivative formulas for the multivariable H-functionrdquo VijnanaParishad Anusandhan Patrika vol 49 no 2 pp 159ndash175 2006

[10] D L Suthar and P Agarwal ldquoGeneralized mittag -leffler func-tion and the multivariable h-function involving the generalizedmellin-barnes contour integralsrdquoCommunications inNumericalAnalysis vol 2017 no 1 pp 25ndash33 2017

[11] D L Suthar B Debalkie and M Andualem ldquoModifiedSaigo fractional integral operators involving multivariableH-function and general class of multivariable polynomialsrdquoAdvances in Difference Equations vol 2019 no 1 article no 2132019

[12] D L Suthar H Habenom and H Tadesse ldquoCertain integralsinvolving aleph function and Wrightrsquos generalized hypergeo-metric functionrdquo Acta Universitatis Apulensis no 52 pp 1ndash102017

[13] D L Suthar G Reddy and B Shimelis ldquoOn Certain UnifiedFractional Integrals Pertaining to Product of Srivastavarsquos Poly-nomials and N-Functionrdquo International Journal of MathematicsTrends and Technology vol 47 no 1 pp 66ndash73 2017

[14] E D Rainville Special Functions The Macmillan CompanyNew York NY USA 1960

10 Journal of Applied Mathematics

[15] G Szego Orthogonal Polynomials American MathematicalSociety Colloquium Publications vol 23 American Mathemati-cal Society Providence RI USA 1959

[16] A Erdersquolyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol I McGraw-Hill NewYork-Toronto-London 1953

[17] V S Kiryakova Generalized Fractional Calculus and Applica-tions vol 301 of Pitman Research Notes in Mathematics UnitedStates with JohnWiley amp Sons Longman Scientific ampTechnicalNew York NY USA 1994

[18] V P Saxena ldquoFormal solution of certain new pair of dualintegral equations involving H-functionsrdquo Proceedings of theNational Academy of Sciences vol A52 no 3 pp 366ndash375 1982

[19] H M Srivastava ldquoA multilinear generating function for theKonhauser sets of biorthogonal polynomials suggested by theLaguerre polynomialsrdquo Pacific Journal of Mathematics vol 117no 1 pp 183ndash191 1985

[20] H M Srivastava and N P Singh ldquoThe integration of certainproducts of the multivariableH-function with a general class ofpolynomialsrdquo Rendiconti del Circolo Matematico di Palermo vol 32 no 2 pp 157ndash187 1983

[21] F Ayant ldquoThe integration of certains products of special func-tions and multivariable Aleph-functionsrdquo International Journalof Mathematics Trends and Technology vol 31 no 2 pp 101ndash1122016

[22] D Kumar R K Saxena and J Ram ldquoFinite integral formulasinvolving aleph functionrdquo Boletim da Sociedade Paranaense deMatematica vol 36 no 1 pp 177ndash193 2018

[23] D L Suthar S Agarwal and D Kumar ldquoCertain integralsinvolving the product of Gaussian hypergeometric function andAleph functionrdquoHonamMathematical Journal vol 41 no 1 pp1ndash17 2019

[24] D L Suthar and B Debalkie ldquoA new class of integral relationinvolving Aleph-functionsrdquo Surveys in Mathematics and itsApplications vol 12 pp 193ndash201 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Journal of Applied Mathematics

[15] G Szego Orthogonal Polynomials American MathematicalSociety Colloquium Publications vol 23 American Mathemati-cal Society Providence RI USA 1959

[16] A Erdersquolyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol I McGraw-Hill NewYork-Toronto-London 1953

[17] V S Kiryakova Generalized Fractional Calculus and Applica-tions vol 301 of Pitman Research Notes in Mathematics UnitedStates with JohnWiley amp Sons Longman Scientific ampTechnicalNew York NY USA 1994

[18] V P Saxena ldquoFormal solution of certain new pair of dualintegral equations involving H-functionsrdquo Proceedings of theNational Academy of Sciences vol A52 no 3 pp 366ndash375 1982

[19] H M Srivastava ldquoA multilinear generating function for theKonhauser sets of biorthogonal polynomials suggested by theLaguerre polynomialsrdquo Pacific Journal of Mathematics vol 117no 1 pp 183ndash191 1985

[20] H M Srivastava and N P Singh ldquoThe integration of certainproducts of the multivariableH-function with a general class ofpolynomialsrdquo Rendiconti del Circolo Matematico di Palermo vol 32 no 2 pp 157ndash187 1983

[21] F Ayant ldquoThe integration of certains products of special func-tions and multivariable Aleph-functionsrdquo International Journalof Mathematics Trends and Technology vol 31 no 2 pp 101ndash1122016

[22] D Kumar R K Saxena and J Ram ldquoFinite integral formulasinvolving aleph functionrdquo Boletim da Sociedade Paranaense deMatematica vol 36 no 1 pp 177ndash193 2018

[23] D L Suthar S Agarwal and D Kumar ldquoCertain integralsinvolving the product of Gaussian hypergeometric function andAleph functionrdquoHonamMathematical Journal vol 41 no 1 pp1ndash17 2019

[24] D L Suthar and B Debalkie ldquoA new class of integral relationinvolving Aleph-functionsrdquo Surveys in Mathematics and itsApplications vol 12 pp 193ndash201 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom