A characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind

Abstract

We show that the only orthogonal polynomials with a generating function of the form $F(xz-\alpha z^2)$ are the ultraspherical, Hermite, and Chebyshev polynomials of the first kind. For special $F$ for which this is the case, we then finish the classification of orthogonal polynomials with more general generating functions $F(xw(z)-R(z))$.

v3: references added. v2: revised as a research article, additional results about logarithmic-type generating functions included. v1: an expository version of the article

References in corpus (2)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Ultraspherical type generating functions for orthogonal polynomials