Some operator identities and \(q\)-series transformation formulas
type: Item
instance of: scholarly article
title: Some operator identities and \(q\)-series transformation formulas
zbMATH Open document ID: 1021.05010
DOI: 10.1016/S0012-365X(02)00626-X
author: Zhi-Guo Liu
published in: Discrete Mathematics
publication date: 2003-05-25T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 33D15
Mathematics Subject Classification ID: 33D05
zbMATH DE Number: 1915539
zbMATH Keywords: operator identity
zbMATH Keywords: \(q\)-series
zbMATH Keywords: Sears' transformation
zbMATH Keywords: Barnes' lemma
zbMATH Keywords: Andrews' identity
zbMATH Keywords: Ramanujan's beta integral
zbMATH Keywords: bilateral series
zbMATH Keywords: transformation formula of \(q\)-series
zbMATH Keywords: \(q\)-exponential operator
reviewed by: Kokichi Sakai
MaRDI profile type: Publication
full work available at URL: s0012-365x(02)00626-x
OpenAlex ID: W2009483292
description: scientific article; zbMATH DE number 1915539