2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/136263We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's ${}_2ϕ_1$ transformation formula and Sears' ${}_3ϕ_2$ 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-Szegö polynomials, or the continuous big $q$-Hermite polynomials.21 pages, to appear in Advances in Applied MathematicsCombinatorics05A30, 33D05, 33D15text