Zonal spherical functions on the complex reflection groups and (\(n+1,m+1\))-hypergeometric functions
type: Item
instance of: scholarly article
title: Zonal spherical functions on the complex reflection groups and (\(n+1,m+1\))-hypergeometric functions
zbMATH Open document ID: 1054.33011
DOI: 10.1016/S0001-8708(03)00092-6
author: Hiroshi Mizukawa
published in: Advances in Mathematics
publication date: 2004-08-06T00:00:00Z
review text: Let \(C_r\wr S_n\) be the wreath product of the cyclic group \(C_r\) and the symmetric group \(S_n\). Then \((C_r\wr S_n, S_n)\) is a Gelfand pair, that is the permutation representation of \(C_r\wr S_n\) on the homogeneous space \((C_r\wr S_n)/S_n\) decomposes into irreducibles without multiplicity. The paper under rewiew studies the spherical functions of this Gelfand pair. The author shows that they can be expressed in terms of multivariated hypergeometric functions called \((n+1,m+1)-\)hypergeometric functions. He also shows a relation between monomial symmetric functions and hypergeometric functions and derives orthogonality relations.
reviewed by: Fabio Scarabotti
Mathematics Subject Classification ID: 33C80
Mathematics Subject Classification ID: 05E35
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 43A90
zbMATH DE Number: 2081953
zbMATH Keywords: hypergeometric function
zbMATH Keywords: finite Gelfand pair
zbMATH Keywords: spherical function
MaRDI profile type: Publication
cites work: Q4222737
cites work: Higher Specht polynomials
cites work: Q3218140
cites work: Q4090662
cites work: Cube group invariant spherical harmonics and Krawtchouk polynomials
cites work: Orthogonal polynomials of several variables
cites work: Q3788294
cites work: Q3772956
cites work: Q3952291
cites work: On hypergeometric functions in several variables. II: The Wronskian of the hypergeometric functions of type \((n+1,m+1)\)
cites work: ON THE RANK OF THE HYPERGEOMETRIC SYSTEM E(n+1, m+1; α)
cites work: Q3947818
cites work: Some q-Krawtchouk Polynomials on Chevalley Groups
cites work: Three addition theorems for some q-Krawtchouk polynomials
cites work: Hypergeometric Functions, My Love
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description: scientific article; zbMATH DE number 2081953