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Some operator identities and \(q\)-series transformation formulas. (English) Zbl 1021.05010

In this paper it is shown 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 if it shown that the two term \({_3\phi_2}\) transformation formula of Sears can be recovered from Heine’s transformation formula, and the celebrated Sears \({_4\phi_3}\) transformation formula can be derived from his \({_3\phi_2}\) transformation formulation with the same method. New proofs of the three term Sears \({_3\phi_2}\) transformation formula and an identity of Andrews are also provided. The \(q\)-analogue of Barnes’ second lemma is re-derived from the \(q\)-analogue of Barnes’ first lemma in one step. In addition, two of Ramanujan’s formulas for beta integrals are generalized to two more general integrals. Finally, two general transformation formulas for bilateral series are established.

MSC:

05A30 \(q\)-calculus and related topics
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
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