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

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 2 F 1 _2F_1 function. Based on a q q -series transformation formula and the Nassrallah–Rahman integral we prove a q q -beta integral which has twelve parameters, with several other results, both classical and new, included as special cases. This q q -beta integral also allows us to derive a curious double q q -series transformation formula, which includes one formula of Al-Salam and Ismail as a special case.