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

TL;DR

This paper characterizes specific polynomial sets with particular generating functions and recursion relations, identifying the monomials and rescaled ultraspherical, Hermite, and Chebyshev polynomials of the first kind as unique solutions.

Contribution

It provides a new characterization theorem that uniquely identifies these classical polynomial families based on their generating functions and three-term recursion relations.

Findings

Only monomials and rescaled ultraspherical, Hermite, Chebyshev polynomials satisfy the given conditions.

The characterization is based on the form of the generating function and recursion relation.

The result narrows the class of polynomial sets with specific generating functions and recursions.

Abstract

We show that the only polynomial sets with a generating function of the form F (xt -- R(t)) and satisfying a three-term recursion relation are the monomial set and the rescaled ultraspherical, Hermite, and Chebyshev polynomials of the first kind.