A note on orthogonal polynomials described by Chebyshev polynomials

TL;DR

This paper extends existing results on orthogonal polynomials associated with a weight function involving Chebyshev polynomials, providing a unified approach for weights of the form |T_m(x)|^p over [-1,1].

Contribution

It introduces a simple, unified method to analyze orthogonal polynomials with weights based on Chebyshev polynomials, generalizing previous results.

Findings

Extended results on orthogonal polynomials with Chebyshev-based weights

Unified approach simplifies analysis of these polynomials

Applicable for weights with p > -1

Abstract

The purpose of this note is to extend in a simple and unified way some results on orthogonal polynomials with respect to the weight function $$\frac{|T_m(x)|^p}{\sqrt{1-x^2}}\;,\quad-1<x<1\;,$$ where $T_m$ is the Chebyshev polynomial of the first kind of degree $m$ and $p>-1$.