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 2F₁ 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.