q Extensions

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J. Math. Anal. Appl. 320 (2006) 642648

www.elsevier.com/locate/jmaa

q-Extensions of certain Erdlyi type integrals

C.M. Joshi a, Yashoverdhan Vyas b,,1

a 106, Arihant Nagar, Kalka Mata Road, Udaipur-313 001, Rajasthan, India

b Department of Mathematics and Statistics, Maharana Bhupal Campus, Mohanlal Sukhadia University,Udaipur-313 001, Rajasthan, India

Received 9 January 2005

Available online 24 August 2005

Submitted by H.M. Srivastava

Abstract

Recently we discovered several new Erdlyi type integrals. In the present paper, it is shown how

theq-extensions of all those integrals involving and representing certain q-hypergeometric functions

can be developed. The well-known special cases and applications of theseq-integrals are also pointed

out.

2005 Elsevier Inc. All rights reserved.

Keywords:Erdlyi type integrals;q -Hypergeometric functions;q-Integrals

1. Introduction

In a recent paper [6], we gave an alternative way of proof for Erdlyis integrals (see [6,

Eqs. (1.3)(1.5)]) using series manipulation technique and classical summation theorems.

Motivated from this way of proof, seven new Erdlyi type integrals (see [6, Eqs. (4.1) and

(3.2)(3.7)]) for certain q+1Fq(z) were conjectured and proved. Their multidimensionalextensions and the integrals for convergent series involving arbitrary bounded sequences

* Corresponding author.

E-mail addresses:[email protected], [email protected] (Y. Vyas).1 The author is grateful to theCouncil of Scientific and Industrial Research, New Delhi, India for providing the

Senior Research Fellowship.

0022-247X/$ see front matter 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmaa.2005.07.030


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C.M. Joshi, Y. Vyas / J. Math. Anal. Appl. 320 (2006) 642648 643

of complex numbers were also developed. Further, the way to obtain Pochhammers double

loop contour analogues of all the Erdlyi type integrals, with relaxed convergence condi-

tions were also pointed out.

Motivated from the above cited work, in this paper, we present the q -extensions of the

Erdlyi type integrals [6, Eqs. (4.1), (3.2)(3.7) and (4.5)]. In Section 2, the brief outline

of the proof of three representative results (1.2), (1.3) and (1.14) is given. Some well-

known special cases and applications of these q-extensions are pointed out in Section 3.

For the sake of precision the multidimensional treatment (see [6, pp. 127 and 133135]

and [7, pp. 287288]) of the Erdlyi type q-integrals is avoided. Further, as a fact, the

Pochhammers contour analogue of Erdlyi typeq -integrals are not derivable, since, as far

as the authors are aware of the literature, no exactq-analogue of the Pochhammers contour

or any other contour has been studied previously (see for instance [2, p. 56, 2.7]). However

if at any stage one is successful in obtaining q-analogue of the Pochhammers contourintegral, one may attempt to develop the Pochhammers contour analogue of Erdlyi type

q-integrals.

One interesting fact about the q-analogues of Erdlyi type integrals is that there may

arise two kinds of q-analogues of one Erdlyi type integral. The first kind of the q-

analogues involve t1 as a numerator parameter in one of the q -hypergeometric functionof the integrand, where t is the variable of integration, while the second kind of the q-

analogues do not involve t1 as a numerator parameter in anyq-hypergeometric functionof the integrand. All the Erdlyi type integrals have the first kind ofq-analogues, while

some of them also possess the second kind of the q-analogues. The proof of second kind

q-analogue of an Erdlyi type integral is similar to its ordinary analogue, whereas the proofof first kind q-analogue of an Erdlyi type integral differs from the proof of its ordinary

analogue by one step, in which the use of the definition ofq-integral as an infinite series

(see [4, 1.11]), viz.,

10

f(t)d q t= (1 q)n=0

fqnqn (1.1)

is followed by the series manipulation [8, p. 100, (1)].

Theq -extensions of the Erdlyi type integrals are as follows:

32

,,

, ;q, z

= q( )

q()q( )

10

t1(tq,tzq+;q)(tz,tq;q)

32 ,

,

, ;q, zt

32

, , t1

, qtz

;q,qdq t, (1.2)

32

,,

, ;q, z

= q( )

q()q( )

10

t1(tq,tzq+;q)(tz,tq;q)

32 ,

,

, ;q, zt

32

, , t, tz

;q, z

dq t,

(1.3)


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644 C.M. Joshi, Y. Vyas / J. Math. Anal. Appl. 320 (2006) 642648

54

,

,

,q

,q

,,q,q ;q,

z

=q( )

q( )q( + )

1

0

t1 (tq,tzq;q)(tz,tq+;q) 2

1

,

;q, zt

76 ,

,,q,q,t,t1

,,q,q,tzq, qtz

;q,qdq t, (1.4)

32,, q

q , q ;q,

z

= q(1+ )q()q(1+ )

1

0

t

1 (tq,tzq

,tzq2

;q)

(tz,tzq+, tq1+;q)

77 , q

,q,,,, tq

,, q

,q ,q ,tzq,0

;q, tzq

dq t,

(1.5)

32

,,

q

q ,q

;q, z= q(1+ )

q()q(1+ )

10

t1 (tq,tzq;q)(tz,tq1+;q)

77 , q,q,,,,

tq

,, q , q

, q

,tzq,0 ;q,

tzq

dq t,

(1.6)

32

,,

q

q ,q

;q, z= q(1+ )

q()q(1+ )

10

t1 (tq,tzq;q)(tz,tq1+;q)

86,q

,q,,,,t 1,0

,, q,q ,q , qtz

;q, tq

dq t,

(1.7)

65

,,

,

,q

,q

,,,q,q ;q,

z

= q( )q()q( )

1

0

t1 (tq,tzq;q)

(tq ,tzq;q) 32

,,t1,tzq

; q, t2qz

dq t, (1.8)

43,,, q

q , q

,

;q,

z

= q(1+ )q()q(1+ )


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1

0

t1 (tq,tzq;q)(tz,tq1+

;q)

32

,

, tq

q

,tz;q, z

109

, q

,q

,,, q,,,q,q

,

,q ,q , ,,,q,q

;q, zt2

dq t,

(1.9)

43

,,,

q

q ,q ,

;q, z

= q(1+ )q()q(1+ )

1

0

t1 (tq,tzq;q)(tz,tq1+;q) 3

2

,

, t1

q

, qtz

;q, q

109

, q

,q

,,, q,,,q,q

,

,q ,q , ,,,q,q

;q,zt2dq t,

(1.10)

76

,,

q

, q,q,q,q

q ,q,,,q,q ;q, z

= q( + )q()q( )

10

t1(tq,qzt2,ztq1+;q)(zt,q t,zt2q1+;q)

77, q

,q,,, zqt2

, t

,, q

,q , zt2,

zqt ,0

;q,zqt

dq t, (1.11)

76

,,

q

, q,q,q,q

q ,q,,,q,q ;q, z

= q( + )q()q( )

10

t1(tq,qzt2,ztq1+;q)(zt,q t,zt2q1+;q)

86,q

,q,,, q

zt2,0, t1

,, q

,q ,

zt2, qzt

;q, t

dq t, (1.12)

87

,,,

q2

, q,q,q,q

q ,q , ,,

,q,

q,

;q, z

= q()q()q( )

10

t1(tq,tzq;q)(tz,tq;q) 3

2

, q

, t1

, tz

;q, t2qz


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646 C.M. Joshi, Y. Vyas / J. Math. Anal. Appl. 320 (2006) 642648

65,q

,q,

, , q

,, q ,q ,

;q, t2z

dq t, (1.13)

n=0

cn(,;q)n(,;q)n

= q()q()q()q()q(+ )

10

t1 (tq;q)

(tq+;q)

31

,, t1

;q, t

(zt)dq t, (1.14)

where(z)is assumed to be a convergent series defined by (z)=n=0 cnzn,wherecnis any bounded sequence of complex numbers.

The q-integrals (1.2), (1.4), (1.7), (1.8), (1.10) and (1.12)(1.14) are the first kind q-

analogues of the Erdlyi type integrals [6, Eqs. (4.1), (3.2)(3.7) and (4.5)], respectively

and the q-integrals (1.3), (1.5), (1.6), (1.9) and (1.11) are the second kindq-analogues ofthe Erdlyi type integrals [6, Eqs. (4.1), (3.3), (3.5), (3.6)], respectively. It may be noted

that the Erdlyi type integrals [6, Eqs. (3.2), (3.4), (3.7) and (4.5)] have no second kind

q-analogue.

2. Proofs of Erdlyi typeq-integrals

To prove (1.2), we write the q-integral on the RHS of (1.2) as an infinite series

(indexed-k), using (1.1) and also the involved 32s as m- and n-series, respectively, to

get

RHS of (1.2)= q( )q()q( )

k=0

m=0

kn=0

(1 q)qkqk(+m1)

(q1+k, zq++k;q)(zqk, q+k;q)

(q, q+, qk;q)n(q, q, q;q)m

(q, q1kz , q;q)n(q, q, q;q)m

(z)mqn

m . (2.1)

Applying series manipulation [8, p. 100, (1)] on the k- and n-series in the above tripleq-series and adjusting the powers ofq and theq -shifted factorials, we express thek-series

again as q-integral, using (1.1). Writing the q-integral as a 21(z), using Thomaes q-

analogue of Eulers integral [4, Eq. (1.11.9)] and then expressing this 21(z)as ak-series

and simplifying, we can obtain one other triple q-series. Now applying the triple series

manipulation [6, p. 128], on this triple q-series and summing the inner n-series by the

q-ChuVandermonde sum [4, p. 236, (II.6)] and then the inner m-series by the q-Pfaff

Saalschtz sum [4, p. 237, (II.12)], we get the LHS of (1.2) and this completes the proof.

The (1.3) can be proved on the line of its ordinary analogue [6, Eq. (4.1)]. Out of the

remainingq -integrals, except (1.14), the first kindq -analogue can be proved on the line of

the proof of (1.2) and the second kind q -analogue on the line of (1.3).

To prove (1.14), we write theq -integral as an infinite series (indexed-k), using (1.1) and

also the involved 31 and (zt)as m-series andn-series and thus we get a triple q -series.


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In this tripleq -series, applying series manipulation [8, p. 100, (1)] on k-series andm-series

and simplifying, we obtain the following tripleq -series:

RHS of (1.14)= q()q()q()q()q(+ )

k,m,n=0cn

(q, q;q)m(q+, q;q)m

znqm2+mn+mqkqk(+n1)

(q1+k;q)

(q++k+m;q). (2.2)

Now, expressing thek -series in turn as the q -integral, we see that the appeared q -integral

is in fact a q-Beta integral. The remaining proof is similar to the ordinary version [6,Eq. (4.5)].

3. Well-known special cases and applications of Erdlyi typeq-integrals

The q-integral (1.2) generalizes and unifies Gaspers results [3, Eqs. (1.3) and (1.4)]

and (1.14) is a generalization of [3, Eq. (1.5)]. Several other well-known special cases of

theq -integrals (1.2)(1.14) can be discussed on the line of their ordinary versions studied

in [6]. Further, the q-integrals (1.2)(1.13) generalize certain q-hypergeometric transfor-

mations and also give rise to some hitherto unrecorded q -hypergeometric transformations.

To deduce aq-hypergeometric transformation from an Erdlyi type q-integral we convert

the q-integral into a triple q-series, in which all the sums run from 0 to . After havingconverted the RHS of an Erdlyi type q -integral into such a triple q -series, if we can take

a 21 as inner series and sum it by the q-Gauss sum [4, p. 236, (II.8)], by particularizing

z, we get a double q-series, whicheithercan be converted into a single q-series by sum-

ming one more inner seriesor can be kept as such according to the situation and thus we

get a q-hypergeometric transformation, provided the convergence conditions of the series

involved are being satisfied. Working in this way, (1.3) converts into a well-known trans-

formation of32 series [4, Eq. (III.9)] and (1.5)(1.7) provide three different q -analoguesof a well-known transformation of a very well poised6F5(1)into a 3F2(1)whereas (1.9)gives rise to a hitherto undiscovered transformation formula, viz.,

109

, q

,q

,,, q,q,q,,

,

,q ,q , ,q

,

q

,

q

,q

;q, q

=(q;q)

(q;q) 4

3

,,,

q

q ,q ,

;q, q2

. (3.1)

Further, working as above the (1.2) leads to a generalization of a 32 transformation [4,

Eq. (III.9)] to six free parameters, given by Eq. (3.2) below. Here it may be noted that a

similar generalization for one other 32 transformation [4, Eq. (III. 10)] was deduced by

Andrews [1].


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