Multidimensional matrix inversions and \(A_r\) and \(D_r\) basic hypergeometric series
type: Item
instance of: scholarly article
title: Multidimensional matrix inversions and \(A_r\) and \(D_r\) basic hypergeometric series
zbMATH Open document ID: 0934.33006
DOI: 10.1023/A:1009705129155
published in: The Ramanujan Journal
publication date: 2000-04-10T00:00:00Z
review text: This is a significant contribution to the theory of multiple basic hypergeometric series associated to root systems. This theory has been developed over the past twenty years by mainly Milne and Gustafson and co-workers. A spectacular breakthrough was achieved recently by \textit{S. C. Milne} and \textit{G. M. Lilly} [Bull. Am. Math. Soc., New Ser. 26, No. 2, 258-263 (1992; Zbl 0757.33008)] by extending the classical Bailey transform to the multidimensional case. Thus, \textit{S. C. Milne, G. M. Lilly} [Discrete Math. 139, No. 1-3, 319-346 (1995; Zbl 0870.33012)]; \textit{S. C. Milne} and \textit{J. W. Newcomb} [J. Comput. Appl. Math. 68, No. 1-2, 239-285 (1996; Zbl 0870.33013)] were able to derive new multidimensional generalizations of classical identities such as Jackson's very-well-poised \(_8\varphi_7\)-summation, Watson's \(_8\varphi_7\) to \(_4\varphi_3\) transformation, etc. As \textit{G. E. Andrews} [in: Proc. Symp. Pure Math., Am. Math. Soc., Columbus, Ohio 1978, Proc. Symp. Pure Math. 34, 1-24 (1979; Zbl 0403.33002)] showed, the Bailey transformation is basically equivalent to a particular matrix inversion. On the other hand, this matrix inversion is just a special case of the reviewer's matrix inversion [Proc. Am. Math. Soc. 124, No. 1, 47-59 (1996; Zbl 0843.15005)]. The latter matrix inversion does have other specializations as well which are important in the theory of (one-dimensional) basic hypergeometric series, in particular the specialization that enabled \textit{G. Gasper} and \textit{M. Rahman} to derive many quadratic, cubic and quartic basic hypergeometric identities [see Chap. 3.6 in the book ``Basic hypergeometric series'' (1990; Zbl 0695.33001)]. The central result of the paper under review is an extension (actually two extensions) to the multivariate setting of the reviewer's matrix inversion. The author proves it by using the operator method of the reviewer from [Trans. Am. Math. Soc. 305, No. 2, 431-465 (1988; Zbl 0653.05007)]. With this …
reviewed by: C. Krattenthaler
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 33C70
Mathematics Subject Classification ID: 33D65
zbMATH DE Number: 1185667
zbMATH Keywords: multidimensional matrix inversions
zbMATH Keywords: \(A_r\) basic hypergeometric series
zbMATH Keywords: \(D_r\) basic hypergeometric series
author: Michael J. Schlosser
MaRDI profile type: Publication
full work available at URL: a:1009705129155
OpenAlex ID: W2122501624
description: scientific article; zbMATH DE number 1185667