## Orthogonal polynomials arising from the wreath products of a dihedral group with a symmetric group.(English)Zbl 1064.33009

Summary: Some classes of orthogonal polynomials are discussed in this paper which are expressed in terms of $$(n+1,m+1)$$-hypergeometric functions. The orthogonality comes from that of zonal spherical functions of certain Gelfand pairs.

### MSC:

 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 05E35 Orthogonal polynomials (combinatorics) (MSC2000) 05E05 Symmetric functions and generalizations
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### References:

 [1] Aomoto, K.; Kita, M., Theory of hypergeometric functions, (1994), Springer Tokyo, (in Japanese) [2] Dunkl, C., A krawtchouk polynomial addition theorem and wreath products of symmetric groups, Indiana univ. math. J., 25, 4, 335-358, (1976) · Zbl 0326.33008 [3] Macdonald, I.G., Symmetric functions and Hall polynomials, (1995), Clarendon Press Oxford · Zbl 0487.20007 [4] H. Mizukawa, Zonal spherical functions of (G(r,1,n),Sn) and (n+1,m+1)-hypergeometric functions, Adv. Math., in press. · Zbl 1054.33011 [5] Yoshida, M., Hypergeometric functions, my love. modular interpretations of configuration spaces, aspects of mathematics, (1997), Friedr. Vieweg and Sohn Braunschweig · Zbl 0889.33008
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