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Generalized hypergeometric functions and Laguerre polynomials in two variables. (English) Zbl 0790.33015
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).
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. 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 $\sb 0{\cal F}\sb 1\sp{(d)}$. For the entire collection see [Zbl 0771.00045].
Reviewer: J.Faraut (Paris)

##### MSC:
 33C70 Other hypergeometric functions and integrals in several variables 33C45 Orthogonal polynomials and functions of hypergeometric type 33C20 Generalized hypergeometric series, ${}_pF_q$