On a Reduction Formula for a Kind of Double q-Integrals

Zhi-Guo Liu

doi:10.3390/sym8060044

Abstract

Using the q-integral representation of Sears’ nonterminating extension of the q-Saalschutz summation, we derive a reduction formula for a kind of double q-integrals. This reduction formula is used to derive a curious double q-integral formula, and also allows us to prove a general q-beta integral formula including the Askey–Wilson integral formula as a special case. Using this double q-integral formula and the theory of q-partial differential equations, we derive a general q-beta integral formula, which includes the Nassrallah–Rahman integral as a special case. Our evaluation does not require the orthogonality relation for the q-Hermite polynomials and the Askey–Wilson integral formula.

Highlights

Using the q-integral representation of Sears’ nonterminating extension of the q-Saalschütz summation, we derive a reduction formula for a kind of double q-integrals
When c = 0, this q-integral formula reduces to the following q-integral formula due to Andrews and Askey [6], which can be derived from Ramanujan 1 ψ1 summation
The main purpose of this paper is to study double q-integrals

Summary

A Double q-Integral Formula

Throughout the paper, we assume that 0 < q < 1. The principal result of this paper is the following general iterated q-integral formula: Theorem 1. On replacing c by y in the Al-Salam–Verma integral in Proposition 2, we immediately find that u dq x = Combining these two equations, we complete the proof of Theorem 1. When r = cduvw, the 3 φ2 series in the above equation reduces to 1, and we have the following curious double q-integral formula. Using Theorem 1, we can prove the following general q-beta integral formula that includes the. Using this theorem and the theory of q-partial differential equations developed recently by us, we can prove the following general q-beta integral formula, which includes the Nassrallah–Rahman integral as a special case.

The Proof of Theorem 2

The Proof of Theorem 3 and the Askey–Wilson Integral

The Rogers–Szegő Polynomials and q-Hermite Polynomials

The Proof of Theorem 4

Conclusions