The relation of the d-orthogonal polynomials to the Appell polynomials

We are dealing with the concept of d-dimensional orthogonal (abbreviated d-orthogonal) polynomials, that is to say polynomials verifying one standard recurrence relation of order d + 1. Among the d-orthogonal polynomials one singles out the natural generalizations of certain classical orthogonal polynomials. In particular, we are concerned, in the present paper, with the solution of the following problem (P): Find all polynomial sequences which are at the same time Appell polynomials and d-orthogonal. The resulting polynomials are a natural extension of the Hermite polynomials.

A sequence of these polynomials is obtained. All the elements of its (d + 1)-order recurrence are explicitly determined. A generating function, a (d + 1)-order differential equation satisfied by each polynomial and a characterization of this sequence through a vectorial functional equation are also given. Among such polynomials one singles out the d-symmetrical ones (Definition 1.7) which are the d-orthogonal polynomials analogous to the Hermite classical ones. When d = 1 (ordinary orthogonality), we meet again the classical orthogonal polynomials of Hermite.