Consequences of the \(A_ l\) and \(C_ l\) Bailey transform and Bailey lemma
type: Item
instance of: scholarly article
title: Consequences of the \(A_ l\) and \(C_ l\) Bailey transform and Bailey lemma
zbMATH Open document ID: 0870.33012
DOI: 10.1016/0012-365X(94)00139-A
author: Stephen C. Milne
author: Glenn M. Lilly
published in: Discrete Mathematics
publication date: 1997-09-08T00:00:00Z
review text: Over the past 15 years, a theory of identities for multiple (basic) hypergeometric series associated to root systems (but not requiring any knowledge of root systems) was built up, by Stephen Milne and his followers. This theory extends the classical theory of (basic) hypergeometric series form single sums to multiple summations. The paper under review constitutes a fundamental corner stone in the development of this theory. It is well-known that the classical Bailey lemma and Bailey transform allow to conveniently derive new identities from known ones. In this paper, two different extensions of the Bailey lemma and transform to the multiple case, one associated to root systems \(A_\ell\), the other associated to root systems \(C_\ell\), are given. Whereas previously proofs of identities in this multiple series theory required usually involved and labourious recursive arguments, these \(A_\ell\) and \(C_\ell \) extensions of the Bailey lemma and transform now allow to produce an host of new identities with ease, including extensions of many important classical identities to the multiple case. In this paper, \(A_\ell\) and \(C_\ell\) extensions of Watson's \(_8\Phi_7\)-to-\(_4\Phi_3\)-transformation, of Jackson's very-well-poised \(_8\Phi_7\)-summation, and of Sears' balanced \(_4\Phi_3\)-transformation are derived. Further applications can be found in [\textit{S. C. Milne}, ``Balanced \(_3\Phi_2\) summation theorems for \(U(n)\) basic hypergeometric series'' preprint; \textit{S. C. Milne} and \textit{J. W. Newcomb}, J. Comput. Appl. Math. 68, No. 1-2, 239-285 (1996; reviewed below)]. More general inversion formulas, which are at the heart of the Bailey lemma, were found by Bhatnagar, Milne, and Schlosser [\textit{G. Bhatnagar} and \textit{S. C. Milne}, ``Generalized bibasic hypergeometric series and their \(U(n)\) extension'', Adv. Math., to appear; \textit{M. Schlosser}, ``Multidimensional matrix inversions and \(A_r\) and \(D_r\) basic hypergeometric series'', Ram…
Mathematics Subject Classification ID: 33D80
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 773916
zbMATH Keywords: basic hypergeometric series
zbMATH Keywords: root systems
zbMATH Keywords: Bailey lemma
zbMATH Keywords: Bailey transform
reviewed by: C. Krattenthaler
MaRDI profile type: Publication
Wikidata QID: Q124839596
cites work: Q3820674
cites work: On Quadratic Transformations of Basic Series
cites work: Q4109912
cites work: Q4146776
cites work: A General Theorem on Partitions with Difference Conditions
cites work: Multiple series Rogers-Ramanujan type identities
cites work: Q3724706
cites work: The Fifth and Seventh Order Mock Theta Functions
cites work: Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities
cites work: Partitions and indefinite quadratic forms.
cites work: Q4760447
cites work: Some Identities in Combinatory Analysis
cites work: Identities of the Rogers-Ramanujan Type
cites work: Q3324796
cites work: A pattern calculus for tensor operators in the unitary groups
cites work: Q3929699
cites work: Q3961690
cites work: Some identities for terminating q-series
cites work: A Matrix Inverse
cites work: Some inverse relations
cites work: Three Identities in Combinatory Analysis
cites work: Summation, Transformation, and Expansion Formulas for Bibasic Series
cites work: Q4040884
cites work: Applications of q-Lagrange Inversion to Basic Hypergeometric Series
cites work: Another family of q-Lagrange inversion formulas
cites work: Summation Theorems for Hypergeometric Series in $U(n)$
cites work: On Hypergeometric Series Well-Poised in $SU(n)$
cites work: The major counting of nonintersecting lattice paths and generating functions for tableaux
cites work: The \(C_ \ell\) Bailey transform and Bailey lemma
cites work: Generalized hypergeometric functions with applications in statistics and physical sciences
cites work: An elementary proof of the Macdonald identities for \(A_{\ell}^{(1)}\)
cites work: Basic hypergeometric series very well-poised in U(n)
cites work: Q3798727
cites work: Q3832839
cites work: Balanced \(_ 3\phi_ 2\) summation theorems for \(U(n)\) basic hypergeometric series
cites work: A \(q\)-analog of a Whipple's transformation for hypergeometric series in \(U(n)\)
cites work: The $\bal $\ and $\bcl $\ Bailey Transform and Lemma
cites work: On identities of the Rogers-Ramanujan type
cites work: A New Proof of Rogers's Transformations of Infinite Series
cites work: Further Identities of the Rogers-Ramanujan Type
cites work: Q5508372
description: scientific article; zbMATH DE number 773916