Chebyshev polynomials and generalized complex numbers

TL;DR

This paper explores the relationship between Chebyshev polynomials, generalized complex numbers, and matrix representations, providing new insights into their connections and extending to two-variable cases and Hermite polynomials.

Contribution

It introduces a novel perspective on Chebyshev polynomials through the lens of generalized complex numbers and matrix representations, including extensions to two-variable and third-order Hermite polynomials.

Findings

Chebyshev polynomials linked to matrix powers and trigonometric functions

Connection established between two-variable Chebyshev polynomials and third-order Hermite polynomials

New matrix-based interpretations of generalized complex numbers

Abstract

The generalized complex numbers can be realized in terms of $2\times2$ or higher-order matrices and can be exploited to get different ways of looking at the trigonometric functions. Since Chebyshev polynomials are linked to the power of matrices and to trigonometric functions, we take the quite natural step to discuss them in the context of the theory of generalized complex numbers. We also briefly discuss the two-variable Chebyshev polynomials and their link with the third-order Hermite polynomials.