Operator identities and several \(U(n+1)\) generalizations of the Kalnins--Miller transformations
type: Item
instance of: scholarly article
title: Operator identities and several \(U(n+1)\) generalizations of the Kalnins--Miller transformations
zbMATH Open document ID: 1113.33020
author: Zhizheng Zhang
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-11-15T00:00:00Z
review text: Multiple basic hypergeometric series associated to the unitary group \(U(n+1)\) have been studied by many authors. In particular, in \textit{S. C. Milne}, [``Balanced \(_ 3\phi_ 2\) summation theorems for \(U(n)\) basic hypergeometric series.'' Adv. Math. 131, No.1, 93-187 (1997; Zbl 0886.33014)], among many other things, several \(U(n+1)\) generalizations of the \(q\)-Chu-Vandermonde summation formula were obtained. In the paper under review, starting fom those summations formulas, the author proves several \(U(n+1)\) generalizations of the Kalnins-Miller transformation (this was introduced, in the one dimensional case, in \textit{E. G. Kalnins, W. Miller, jun.} [``\(q\)-series and orthogonal polynomials associated with Barnes' first lemma.'' SIAM J. Math. Anal. 19, No.5, 1216-1231 (1988; Zbl 0652.33007)] and studied in \textit{Z.-G. Liu}, [``Some operator identities and \(q\)-series transformation formulas.'' Discrete Math. 265, No.1-3, 119-139 (2003; Zbl 1021.05010)]. The author makes also use of the \(q\)-exponential operator technique, introduced in \textit{W. Y. C. Chen, Z.-G. Liu}, [``Parameter augmentation for basic hypergeometric series. I.'' Mathematical essays in honor of Gian-Carlo Rota's 65th birthday. Boston, MA: Birkhäuser. Prog. Math. 161, 111-129 (1998; Zbl 0901.33008)] and \textit{W. Y. C. Chen, Z.-G. Liu}, [``Parameter augmentation for basic hypergeometric series. II.'' J. Comb. Theory, Ser. A 80, No.2, 175-195 (1997; Zbl 0901.33009)].
reviewed by: Fabio Scarabotti
Mathematics Subject Classification ID: 33D67
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 33D15
zbMATH DE Number: 5073013
zbMATH Keywords: basic hypergeometric series
zbMATH Keywords: multiple basic hypergeometric series
zbMATH Keywords: Kalnins-Miller transformation
zbMATH Keywords: Chu-Vandermonde summation
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.12.073
OpenAlex ID: W2071753892
cites work: Q3724706
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cites work: Some operator identities and \(q\)-series transformation formulas
cites work: Balanced \(_ 3\phi_ 2\) summation theorems for \(U(n)\) basic hypergeometric series
cites work: Two operator identities and their applications to terminating basic hypergeometric series and \(q\)-integrals
cites work: Applications of operator identities to the multiple \(q\)-binomial theorem and \(q\)-Gauss summation theorem
DOI: 10.1016/J.JMAA.2005.12.073
description: scientific article; zbMATH DE number 5073013