Zonal spherical functions on the complex reflection groups and ( n+1, m+1)-hypergeometric functions

Abstract

The pair of groups, complex reflection group G( r,1, n) and symmetric group S n , is a Gelfand pair. Its zonal spherical functions are expressed in terms of multivariate hypergeometric functions called ( n+1, m+1)-hypergeometric functions. Since the zonal spherical functions have orthogonality, they form discrete orthogonal polynomials. Also shown is a relation between monomial symmetric functions and the ( n+1, m+1)-hypergeometric functions.