Orthogonal Polynomials for Modified Chebyshev Measure of the First Kind

Abstract

Given numbers ${n,s \in \mathbb{N}}$, ${n \geq 2}$, and the ${n}$th-degree monic Chebyshev polynomial of the first kind ${\widehat T_n(x)}$, the polynomial system “induced” by ${\widehat T_n(x)}$ is the system of orthogonal polynomials ${\{p_{k}^{n,s} \}}$ corresponding to the modified measure ${d \sigma^{n,s}(x)=\widehat T^{2s}_n(x) d\sigma(x)}$, where ${d\sigma(x)=1/\sqrt{1-x^{2}}dx}$ is the Chebyshev measure of the first kind. Here we are concerned with the problem of determining the coefficients in the three-term recurrence relation for the polynomials ${p^{n,s}_{k}}$. The desired coefficients are obtained analytically in a closed form.