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

TL;DR

This paper derives a new twelve-parameter q-beta integral and a double q-series transformation formula using Askey--Wilson polynomials and Nassrallah--Rahman integral, unifying and extending classical results.

Contribution

It introduces a novel twelve-parameter q-beta integral and a double q-series transformation formula, expanding the theoretical framework of q-series and orthogonal polynomials.

Findings

Derived a twelve-parameter q-beta integral.

Established a new double q-series transformation formula.

Unified several classical results as special cases.

Abstract

The Askey--Wilson polynomials are the most general classical orthogonal polynomials that are known and the Nassrallah--Rahman integral is a very general extension of Euler's integral representation of the classical $_2F_1$ function. Based on a $q$-series transformation formula and the Nassrallah--Rahman integral we prove a $q$--beta integral which has twelve parameters, with several other results, both classical and new, included as special cases. This $q$-beta integral also allows us to derive a curious double $q$--series transformation formula, which includes one formula of Al--Salam and Ismail as a special case