Derivations and identities for Chebyshev polynomials of the first and second kinds

TL;DR

This paper develops a unified algebraic framework using derivations to derive new identities for Chebyshev polynomials of both kinds, connecting them with hypergeometric functions and other polynomial families.

Contribution

It introduces Chebyshev derivations and their kernels, providing a novel method to generate polynomial identities for Chebyshev and related polynomials.

Findings

Derived new identities for Chebyshev polynomials of both kinds.

Connected Chebyshev polynomials with hypergeometric functions.

Identified kernels of Chebyshev derivations leading to polynomial identities.

Abstract

In this paper we follow the general approach, proposed earlier by the first author, which is derived from the invariant theory field and provides a way of obtaining of the polynomial identities for any arbitrary polynomial family. We introduce the notion of Chebyshev derivations of the first and second kinds, which is based on the polynomial algebra, and corresponding specific differential operators. We derive the elements of their kernels and prove that any element of the kernel of the derivations defines a polynomial identity satisfied by the Chebyshev polynomials of the first and second kinds. Combining elementary methods and combinatorial techniques, we obtain several new polynomial identities involving the Chebyshev polynomials of the both kinds and a special case of the Jacobi polynomials. Using the properties of the generalised hypergeometric function, we specify the Chebyshev polynomials of the first and second kinds via the generalised hypergeometric function and, as a consequence, derive the corresponding identities involving the generalised hypergeometric function and the Chebyshev polynomials of the first and second kinds.