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

Abstract

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. 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.

The Askey–Wilson Integral and Extensions

“…Ito-Witte [32] made extensions by generalizing the weight functions for the integrals and resolving linear q-difference equations. Liu [33] examined double q-integrals related to the Askey-Wilson integral through the series rearrangement. Szablowski [34] generalized Askey-Wilson integrals by expanding the Askey-Wilson density into continuous q-Hermite polynomials.…”

A note on fractional Askey--Wilson integrals

“…The treatment from the point view of the q-calculus can open new perspectives as it did, for example, in optimal control problems [7,8,15,27]. For more information, see details in [10,11,2,3,21,29,30,31,32,33,24,25,26].…”

Askey–Wilson polynomials and a double $q$-series transformation formula with twelve parameters

“…There are several different ways of proving the Askey-Wilson q-beta integral formula, see, for example [3,4,7,11,13,14,20,23,25].…”