Some remarks about q-Chebyshev polynomials and q-Catalan numbers and related results

TL;DR

This paper surveys generalizations and q-analogues of classical results relating Chebyshev, Lucas, Fibonacci polynomials, and their moments to binomial coefficients and Catalan numbers.

Contribution

It introduces new q-analogues and generalizations of known polynomial moments, expanding the understanding of these classical sequences.

Findings

Presented q-analogues of polynomial moments

Connected Chebyshev, Lucas, Fibonacci polynomials to combinatorial numbers

Extended classical results to a broader algebraic framework

Abstract

The moments of the Lucas polynomials and of the Chebyshev polynomials of the first kind are (multiples of) central binomial coefficients and the moments of the Fibonacci polynomials and of the Chebyshev polynomials of the second kind are Catalan numbers. In this survey paper we present some generalizations of these results together with various q-analogues.