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\((n+1,m+1)\)-hypergeometric functions associated to character algebras. (English) Zbl 1059.33020
The notion of character algebra was introduced by Y. Kawada in [Proc. Phys.-Math. Soc. Japan, III. Ser. 24, 97–109 (1942; Zbl 0063.03172)] and it may be seen as a generalization of the notion of Bose Mesner algebra of a commutative association scheme. In the paper under review, the authors show how to obtain discrete orthogonal polynomials form the eigenmatrices of character algebras. In particular, they generalize the well known relation between Hamming scheme and Krawtchouk polynomials by considering the symmetric tensor spaces of a character algebra. In such a way they obtain polynomials expressed in terms of \((n+1,m+1)\)-hypergeometric functions. The results in this paper generalize those obtained by the first author in [Adv. Math. 184, No. 1, 1–17 (2004; Zbl 1054.33011)].

MSC:
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
05E30 Association schemes, strongly regular graphs
05E35 Orthogonal polynomials (combinatorics) (MSC2000)
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