Orthogonal polynomials arising from the wreath products of a dihedral group with a symmetric group

Abstract

Some classes of orthogonal polynomials are discussed in this paper which are expressed in terms of ðn þ 1; m þ 1Þ-hypergeometric functions. The orthogonality comes from that of zonal spherical functions of certain Gelfand pairs. r

(𝑛+1,𝑚+1)-hypergeometric functions associated to character algebras

“…Also, Akazawa and Mizukawa [1] obtained a similar result for the pair (D(r, n), D (1, n)), where D(r, n) = D r S n is the wreath product of the dihedral group of order 2r and S n . Since the Hecke algebra of any Gelfand pair is a character algebra, our result includes all the cases dealt with in [10,1] (see Remark 2.1).…”

“…If A is the Hecke algebra of a Gelfand pair (G, H), then S n (A) coincides with that of the Gelfand pair (G S n , H S n ) (see [10,1]). In particular, when G = S q and H = S q−1 , S n (A) is the Bose-Mesner algebra of the Hamming association scheme H(n, q).…”

Generalized characters of the symmetric group

“…The theory of characters can be reformulated in terms of Gelfand pairs, see [10,VII,§1]. Specifically, let G be a finite group, and K be a subgroup of G. Denote by C(G, K) the algebra of complex-valued functions f on G (with convolution as the multiplication) such that f (kxk ) = f (x) for all x ∈ G and k, k ∈ K. If C(G, K) is commutative, the pair (G, K) is called a Gelfand pair, and one can associate with (G, K) the set of zonal spherical functions.…”

Zonal polynomials for wreath products

“…Multivariate orthogonal polynomials of hypergeometric type are considered in [1,8,9]. These papers are devoted to multivariate orthogonal polynomials which are expressed in terms of the (m + 1, n + 1)-hypergeometric functions where m = (m 0 , m 1 , .…”

“…[1,2,[8][9][10]) written about Gelfand pairs of wreath products. In this paper, we generalize the theory of the Gelfand pair (S 2n , H n ) to wreath products.…”