The Cauchy operator for basic hypergeometric series

Abstract

We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's ϕ 1 2 transformation formula and Sears' ϕ 2 3 transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator T ( b D q ) . Using this operator, we obtain extensions of the Askey–Wilson integral, the Askey–Roy integral, Sears' two-term summation formula, as well as the q-analogs of Barnes' lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate Rogers–Szegö polynomials, or the continuous big q-Hermite polynomials.