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 has been defined through zonal polynomial expansions. Since zonal polynomials is a partition) are special cases of Jack polynomials
one defines more generally hypergeometric functions in terms of the Jack polynomials. In this paper one studies in detail the case of two variables . Then the Jack polynomials can be expressed in terms of the Jacobi polynomials
, . By using an integral representation of the Jacobi polynomials one is able to express the generalized hypergeometric kernel functions and in terms of the classical hypergeometric functions in one variable.
The generalized Laplace transform with kernel is proved to be injective, and the Laplace transform of is computed. One defines the generalized Laguerre polynomials , establishes a generating formula, an integral representation, and one proves that the set of Laguerre polynomials is an orthogonal basis for a Hilbert space . Finally a generalized Tricomi theorem is proved for the generalized Hankel transform with kernel .