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

TL;DR

This paper characterizes the unique orthogonal polynomials—ultraspherical, Hermite, and Chebyshev of the first kind—that have generating functions of a specific quadratic form, completing their classification under more general conditions.

Contribution

It identifies the only orthogonal polynomials with generating functions of the form F(xz - αz^2) and extends the classification to more general generating functions.

Findings

Ultraspherical, Hermite, and Chebyshev polynomials are uniquely characterized by their generating functions.

The classification of orthogonal polynomials with broader generating functions is completed.

The results provide a complete description of polynomials with specific quadratic generating functions.

Abstract

We show that the only orthogonal polynomials with a generating function of the form $F(x z - \alpha z^2)$ 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 $F(x w(z) - R(z))$.