We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's 2 φ 1 transformation formula and Sears' 3 φ 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 qanalogs 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.

Chu has recently shown that the Abel lemma on summations by parts can serve as the underlying relation for Bailey's 6 ψ 6 bilateral summation formula. In other words, the Abel lemma spells out the telescoping nature of the 6 ψ 6 sum. We present a systematic approach to compute Abel pairs for bilateral and unilateral basic hypergeometric summation formulas by using the q-Gosper algorithm. It is demonstrated that Abel pairs can be derived from Gosper pairs. This approach applies to many classical summation formulas.

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