Abstract

We use the -Chu-Vandermonde formula and transformation technique to derive a more general -integral equation given by Gasper and Rahman, which involves the Cauchy polynomial. In addition, some applications of the general formula are presented in this paper.

1. Introduction and Main Result

It is well known that the -integral is an important branch of -series theory. There are many techniques to achieve the ends; for instance, combinatorics method (cf. [1]), analysis methods (cf. [2–4]), and method of transformation (cf. [5–7]) are usually used. In 1989, Gasper and Rahman applied some analysis techniques to derive the following -contour integral formula (cf. [8, Equation (3.17)]): Inspired by [7, 8], we employ the above equation and transformation technique to derive a more general -contour integral equation. The main result of this paper is stated as follows.

Theorem 1. If , , , , and are nonnegative integers and , then provided that and is a deformation of the unit circle so that the poles of lie outside the contour and the origin and the poles of lie inside the contour. Where denotes the Cauchy polynomial defined as (7), one denotes that , and when , one sets one .

2. Notations and Lemmas

We adopt the custom notations given in [9]. It is supposed that in this paper. We use to denote the set of all nonnegative integers.

For any complex parameter , the -shifted factorials are defined as For brevity, we also use the following notation: The -binomial coefficient and the -binomial theorem are given by The basic hypergeometric series  is given by In this paper, we denote that and , , , , .

Let be any complex variables; then, the Cauchy polynomial is defined as

Recall that -Chu-Vandermonde’s identity (cf. [9, page 14, Equation (1.5.3)]) is given as follows: As we know, it is one of the fundamental formulas in the basic hypergeometric series. Some applications of it were introduced in [5, 10, 11]. We will apply this identity to start our proof in the following. Since we assume that the integrals are the same established condition as the theorem, we omit the condition in the following.

Lemma 2. One has

Proof. We rewrite (8) as follows: Replacing by , respectively, we have
Both sides of (11) multiply by Then, we have Taking the -integral on both sides of (13) with respect to variable , we get
Employing (1) to the left side of (14), we have the desired result after some simplification.

On the other hand, if we multiply (13) by , we have

Taking the -integral on both sides of (15) with respect to variable , we use (9) in the resulting equation. After simple rearrangements, noting that , we get the following.

Lemma 3. One has

Both sides of (11) multiply by Then, taking the -integral on both sides of the result equation with respect to variable , we find the following.

Lemma 4. On has where denote , respectively.

3. Proof and Some Applications

Now, we return to the proof of Theorem 1.

The following result can be easily derived from (16) and (18):

Letting , , and and combining (19) with (18), by induction, similar proof can be performed to get the desired result.

Taking in (2), the theorem goes back to formula (1). Putting in (2), we have the following.

Corollary 5. One has

Letting in (2), we get the following.

Corollary 6. One has

Combining (21) with (18), by induction and applying (2), we can conclude the following.

Theorem 7. One has

Comparing (2) and (22), we have the following interesting identity.

Corollary 8. If , , , , and are nonnegative integers, then

Taking and in (23), we have Setting , then letting in the above identity, we have the following.

Corollary 9. If , , then where and , , .

Taking and in (23), we have Setting , then letting in the above identity, we have the following.

Corollary 10. If , , , then where and , , .

More general, we have the following identity.

Corollary 11. If , , , , , then where , .

Both sides of (20) multiply by ; then, summing from to and using the -binomial theorem, we find the following.

Corollary 12. If , then

Remark 13. If , identity (23) becomes the -Chu-Vandermonde formula.

Acknowledgments

The author would like to thank the referees and the editors for their many valuable comments and suggestions. The author would also like to thank Professor Bruce C. Berndt for his help and warm hospitality accorded to him during his visit to UIUC. The author is, moreover, supported by Jiangsu Overseas Research and Training Program for University Prominent Young and Middle-Aged Teachers and Presidents. The author is also supported by the National Natural Science Foundation of China (no. 10971078).