\(q\)-differential operator identities and applications
type: Item
instance of: scholarly article
title: \(q\)-differential operator identities and applications
zbMATH Open document ID: 1114.33023
author: Jian-Ping Fang
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-06-06T00:00:00Z
review text: The author defines the \(q\)-exponential operator \[ _{1}\Phi_{0}\left(\begin{aligned} b \\ -\, \end{aligned}; q,\,-c\theta\right) =\sum_{n=0}^{\infty}\frac{(b;q)_{n}\,(-c\theta)^{n}}{(q;q)_{n}}\,, \] and gives some of its properties such as transformation formulas and expressions in terms of basic hypergeometric series of \(_{3}\Phi_{2}\) and \(_{2}\Phi_{1}\) type. The author gives a \(_{2}\Phi_{2}\) transformation formula which contains Jackson's \(_{2}\Phi_{2}\) transformation as a special case. The author also gives a formal extension of Bailey's \(_{3}\psi_{3}\) summation formula for bilateral basic hypergeometric series and describes a method of deriving a generalized formula of the Sears three terms of \(_{3}\Phi_{2}\) series transformation. An extension of the Sears terminating balanced \(_{4}\Phi_{3}\) transformation formula as well as a formal extension of Heine's \(_{2}\Phi_{1}\) transformation formula are also given.
reviewed by: Stamatis Koumandos
Mathematics Subject Classification ID: 33D15
Mathematics Subject Classification ID: 33D90
Mathematics Subject Classification ID: 33D45
Mathematics Subject Classification ID: 05A30
zbMATH DE Number: 5161783
zbMATH Keywords: Basic hypergeometric series
zbMATH Keywords: Jackson's summation formula
zbMATH Keywords: bilateral basic hypergeometric series
zbMATH Keywords: transformation formulae
zbMATH Keywords: \(q\)-exponential operator
zbMATH Keywords: \(q\)-differential operator
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.10.087
OpenAlex ID: W1982180298
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DOI: 10.1016/J.JMAA.2006.10.087
description: scientific article; zbMATH DE number 5161783