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Zonal spherical functions on the complex reflection groups and (\(n+1,m+1\))-hypergeometric functions. (English) Zbl 1054.33011
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.

MSC:
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
05E35 Orthogonal polynomials (combinatorics) (MSC2000)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
43A90 Harmonic analysis and spherical functions
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