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.KeywordsExplicit FormulaOrthogonal PolynomialPolynomial SequenceRecurrence CoefficientSymmetric Tridiagonal MatrixThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.