On the classical {$d$}-orthogonal polynomials defined by certain generating functions. II

Abstract

The purpose of this work is to present some results on the d-orthogonal polynomials dened by generating functions of certain forms to be specied below. The resulting polynomials are natural extensions of some classical orthogonal polynomials. The rst part of this study is motived by the recent work of Von Bachhaus [21] who showed that, among the orthogonal polynomials, only the Hermite and the Gegenbauer polynomials are dened by the generating function G 2xt t 2 . Here we generalize this result in the context of d-orthogonality, by considering the polynomials generated by G (d +1)xt t d+1 ,w hered is a positive integer. We obtain that the resulting polynomials are d-symmetric Denition 1.2 and \classical in the Hahn’s sense. We provide some examples to illustrate the results obtained and show that they involve certain known polynomials. Finally, we conclude by giving some properties of the zeros of these polynomials as well as a (d +1 )-order dierential equation satised by each polynomial. In forthcoming paper [2] we will consider the polynomials generated by e t (xt).