Abstract and Applied Analysis
Volume 2013, Issue 1 840704
Research Article
Open Access

Note on a q-Contour Integral Formula of Gasper-Rahman

Jian-Ping Fang, 

Corresponding Author

Jian-Ping Fang

School of Mathematical Science, Huaiyin Normal University, Huai’an, Jiangsu 223300, China hytc.edu.cn

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First published: 10 September 2013
https://doi.org/10.1155/2013/840704
Academic Editor: Shanhe Wu

Abstract

We use the q-Chu-Vandermonde formula and transformation technique to derive a more general q-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 q-integral is an important branch of q-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 q-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 q-contour integral equation. The main result of this paper is stated as follows.

Theorem 1. If m0, m1, …, mr, and h are nonnegative integers and , then

()
provided that |γ/α| < 1 and C is a deformation of the unit circle so that the poles of 1/(az, bz; q) ∞ lie outside the contour and the origin and the poles of 1/(α/z; q) ∞ lie inside the contour. Where Pn(a; b) denotes the Cauchy polynomial defined as (7), one denotes that , and when i = 0, one sets one .

2. Notations and Lemmas

We adopt the custom notations given in [9]. It is supposed that 0<|q | < 1 in this paper. We use N to denote the set of all nonnegative integers.

For any complex parameter a, the q-shifted factorials are defined as
()
For brevity, we also use the following notation:
()
The q-binomial coefficient and the q-binomial theorem are given by
()
The basic hypergeometric series  sΦt  is given by
()
In this paper, we denote that and k, m, n, s, t ∈ N.
Let a, b be any complex variables; then, the Cauchy polynomial Pn(a; b) is defined as
()
Recall that q-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 (a, c) by (α/z, fα), respectively, we have
()

Both sides of (11) multiply by

()
Then, we have
()
Taking the q-integral on both sides of (13) with respect to variable z, 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 q-integral on both sides of (15) with respect to variable z, 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 q-integral on both sides of the result equation with respect to variable z, we find the following.

Lemma 4. On has

()
where (nh+1, dh+1) denote (n, f), 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 n = n0, k = k0, and f = d0 and combining (19) with (18), by induction, similar proof can be performed to get the desired result.

Taking n1 = n2 = ⋯ = nh+1 = 0 in (2), the theorem goes back to formula (1). Putting n1 = ⋯ = nh = 0 in (2), we have the following.

Corollary 5. One has

()

Letting n2 = ⋯ = nh = 0 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 m0, m1, …, mr, and h are nonnegative integers, then

()

Taking h = 1 and d0 = d1 = qb in (23), we have
()
Setting bα = q, then letting q → 1 in the above identity, we have the following.

Corollary 9. If n0, n1 ∈ N, then

()
where (a) 0 = 1 and (a) n = a(a + 1)⋯(a + n − 1), n ≥ 1, n ∈ N.

Taking h = 2 and d0 = d1 = d2 = qb in (23), we have
()
Setting bα = q, then letting q → 1 in the above identity, we have the following.

Corollary 10. If n0, n1, n2 ∈ N, then

()
where (a) 0 = 1 and (a) n = a(a + 1)⋯(a + n − 1), n ≥ 1, n ∈ N.

More general, we have the following identity.

Corollary 11. If h, n0, n1, …, nh ∈ N, then

()
where 0 ≤ ki ≤ ni, i = 0, …, h.

Both sides of (20) multiply by 1/(q; q) n; then, summing n from 0 to ∞ and using the q-binomial theorem, we find the following.

Corollary 12. If max {|1/z|, |b|} < 1, then

()

Remark 13. If n1 = n2 = ⋯ = nh = 0, identity (23) becomes the q-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).

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