A new characterization of d-symmetric clasical d-orthogonal polynomials

What is it about?

We have shown in a previous paper that the symmetric clasical orthogonal polynomials (Ultraspherical, Hermite and Chebyshev polynomials of the first kind) are the only orthogonal polynomials with a generating function of the form F(xt-R(t)), where F(z) and R(z) are formal power series. In this paper, we extend this result to d-orthogonality. Indeed, we show that the only d-symmetric d-orthogonal polynomials generated by F(xt-R(t)) are the d-symmetric clasical d-orthogonal polynomials. We note that this result is a consequence of our work, since we allowed the d-symmetric polynomial set to satisfy a (d+1)-order recursion without being d-orthogonal.

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