Utility of integral representations for basic hypergeometric functions and orthogonal polynomials

We describe the utility of integral representations for sums basic hypergeometric functions. In particular we use these to derive an infinite sequence transformations symmetrizations over certain variables which functions possess. These were studied by Bailey, Slater, Askey, Roy, Gasper and Rahman also used facilitate computation outstanding problems in theory orthogonal polynomials q-Askey scheme. generalize give consequences transformation formulas some fundamental integrals connected nonterminating series Askey–Wilson polynomials. express a ratio q-shifted factorials as symmetric sum two with argument q. The result is then expressed q-integral. Examples applied derivation generating are given well missing function continuous dual q-Hahn