Some operator identities and q-series transformation formulas

Abstract

In this paper, we show how to use the q-exponential operator techniques to derive a transformation formula for the q-Hahn polynomials from the q-Chu–Vandermonde identity. With the same method we also show that the two terms 3 φ 2 transformation formula of Sears can be recovered from Rogers’ iteration of Heine's transformation formula, and the celebrated Sears 4 φ 3 transformation formula can be derived from his 3 φ 2 transformation formula with the same method. We also provide new proofs of the three terms Sears 3 φ 2 transformation formula and an identity of Andrews by our method. We re-derive the q-analogue of Barnes’ second lemma from the q-analogue of Barnes’ first lemma in one step. In addition we generalize two Ramanujan's formulas for beta integrals as two more general integrals . Finally, we establish two general transformation formulas for bilateral series.