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Zbl 0863.33007
Douak, Khalfa
The relation of the $d$-orthogonal polynomials to the Appell polynomials. (English)
[J] J. Comput. Appl. Math. 70, No.2, 279-295 (1996). ISSN 0377-0427

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 $(d+1)$-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.
[V.L.Deshpande (Amalner)]

MSC 2000:: *33C45 Orthogonal polynomials and functions of hypergeometric type
42C05 General theory of orthogonal functions and polynomials

Keywords: Hermite polynomials; generating functions; $d$-orthogonal polynomials; Appell polynomials

Cited in: Zbl 1119.42009

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