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 3φ 2 transformation formula of Sears can be recovered from Rogers' iteration of Heine's transformation formula, and the celebrated Sears 4φ 3 transformation formula can be derived from his 3φ 2 transformation formula with the same method. We also provide new proofs of the three terms Sears 3φ 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. © 2002 Elsevier Science B.V.