Construction and Computation of a New Set of Orthogonal Polynomials

Abstract

In this paper we will discuss how to construct and compute a new set of orthogonal polynomials from an existing one. For a given pair of positive integers (n, r) and a given positive measure dσ(t), we will construct a set of orthogonal polynomials corresponding to the modified measure $d\hat \sigma \left( t \right) = {\left( {{\pi _n}\left( t \right)} \right)^{2r}}d\sigma \left( t \right)$. For r = 2 and the first-kind Chebyshev measure we are able to find explicit formulas for the recurrence coefficients of the new set of polynomials. A conjecture is made on those coefficients for any positive integer r and the first-kind Chebyshev measure. For r = 2 and arbitrary measures, a computational method is proposed. Other results are also stated.