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Zbl 0790.33015
Yan, Zhimin
Generalized hypergeometric functions and Laguerre polynomials in two variables. (English)
[A] Richards, Donald St. P. (ed.), Hypergeometric functions on domains of positivity, Jack polynomials, and applications. Proceedings of an AMS special session held March 22-23, 1991 in Tampa, FL, USA. Providence, RI: American Mathematical Society. Contemp. Math. 138, 239-259 (1992). ISBN 0-8218-5159-4

In statistical analysis the multivariate hypergeometric function $\sb pF\sb q$ has been defined through zonal polynomial expansions. Since zonal polynomials $C\sb \kappa$ $(\kappa$ is a partition) are special cases of Jack polynomials $$J\sb \kappa\left(x\sb 1,\dots,x\sb r;{2\over d}\right)=c\sb \kappa C\sp{(d)}\sb \kappa(x\sb 1,\dots,x\sb r)$$ $(C\sb \kappa=C\sb \kappa\sp{(1)})$ one defines more generally hypergeometric functions $\sb pF\sb q\sp{(d)}$ in terms of the Jack polynomials. In this paper one studies in detail the case of two variables $(r=2)$. Then the Jack polynomials can be expressed in terms of the Jacobi polynomials $$C\sb \kappa\sp{(d)}(re\sp t,re\sp{- t})=c(\kappa,d)r\sp{\vert\kappa\vert} P\sp{(\gamma)}\sb{k\sb 1-k\sb 2}(\text{ch} t),\ \gamma={1\over 2}(d-1)$$ $(\kappa=(k\sb 1,k\sb 2)$, $\vert\kappa\vert=k\sb 1+k\sb 2)$. By using an integral representation of the Jacobi polynomials one is able to express the generalized hypergeometric kernel functions $\sb 0{\cal F}\sb 0\sp{(d)}$ and $\sb 0{\cal F}\sb 1\sp{(d)}$ in terms of the classical hypergeometric functions in one variable.\par The generalized Laplace transform with kernel $\sb 0{\cal F}\sb 0\sp{(d)}$ is proved to be injective, and the Laplace transform of $\sb p{\cal F}\sb q\sp{(d)}$ is computed. One defines the generalized Laguerre polynomials $L\sp \gamma\sb \kappa(\gamma\in\bbfR)$, establishes a generating formula, an integral representation, and one proves that the set $\{L\sp \gamma\sb \kappa\}$ of Laguerre polynomials is an orthogonal basis for a Hilbert space $L\sp 2\sb \gamma(\bbfR\sp 2\sb +)$. Finally a generalized Tricomi theorem is proved for the generalized Hankel transform with kernel $\sb 0{\cal F}\sb 1\sp{(d)}$.
[J.Faraut (Paris)]

MSC 2000:: *33C70 Other hypergeometric functions and integrals in several variables
33C45 Orthogonal polynomials and functions of hypergeometric type
33C20 Generalized hypergeometric series

Keywords: zonal polynomials; Jack polynomials; Jacobi polynomials; Laguerre polynomials

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