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The Cauchy Operator for Basic Hypergeometric Series

Abstract

We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's 2ϕ1{}_2\phi_1 transformation formula and Sears' 3ϕ2{}_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(bDq)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 qq-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 qq-Hermite polynomials.Comment: 21 pages, to appear in Advances in Applied Mathematic

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