We introduce the Cauchy augmentation operator for basic hypergeometric
series. Heine's 2ϕ1 transformation formula and Sears' 3ϕ2
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). 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.Comment: 21 pages, to appear in Advances in Applied Mathematic