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 ${}_2{F}_1$ 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