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The relation of the $d$-orthogonal polynomials to the Appell polynomials. (English) Zbl 0863.33007
Author’s abstract: 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 $\left(d+1\right)$-order recurrence are explicitly determined. A generating function, a $\left(d+1\right)$-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.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 42C05 General theory of orthogonal functions and polynomials