The Cauchy Operator for Basic Hypergeometric Series

TL;DR

The paper introduces the Cauchy augmentation operator for basic hypergeometric series, enabling simplified derivations of transformation formulas and extensions of classical integrals and summation formulas, with applications to bivariate polynomials.

Contribution

It presents a new two-parameter Cauchy operator that generalizes existing operators and facilitates derivations of key hypergeometric identities and integrals.

Findings

Derived new transformation formulas for hypergeometric series.

Extended classical integrals like Askey-Wilson and Barnes' lemmas.

Applied the operator to study bivariate Rogers-Szeg"o and big q-Hermite polynomials.

Abstract

We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's ${}_2\phi_1$ transformation formula and Sears' ${}_3\phi_2$ 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 $T(bD_q)$. Using this operator, we obtain extensions of the Askey-Wilson integral, the Askey-Roy integral, Sears' two-term summation formula, as well as the $q$-analogues of Barnes' lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate Rogers-Szeg\"o polynomials, or the continuous big $q$-Hermite polynomials.