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Zbl 1032.33010
Macdonald, Ian G.
Orthogonal polynomials associated with root systems. (English)
[J] Sémin. Lothar. Comb. 45, B45a, 40 p. (2000). ISSN 1286-4889/e

Summary: Let $R$ and $S$ be two irreducible root systems spanning the same vector space and having the same Weyl group $W$, such that $S$ (but not necessarily $R$) is reduced. For each such pair $(R,S)$ we construct a family of $W$-invariant orthogonal polynomials in several variables, whose coefficients are rational functions of parameters $q,t_1,t_2,\dots,t_r$, where $r$ $(= 1, 2$ or 3) is the number of $W$-orbits in $R$. For particular values of these parameters, these polynomials give the values of zonal spherical functions on real and $p$-adic symmetric spaces. Also when $R=S$ is of type $A_n$, they conincide with the symmetric polynomials [described in {\it I. G. Macdonald}, Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press (1995; Zbl 0824.05059), Chapter VI].

MSC 2000:: *33C80 Connections of theory of special functions with groups and algebras
05E05 Symmetric functions
33C52 Orthogonal polynomials and functions associated with root systems

Citations: Zbl 0824.05059

Cited in: Zbl 1132.33012 Zbl 1132.33011

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