Orthogonal polynomials and generalized Gauss-Rys quadrature formulae

Abstract

Orthogonal polynomials and the corresponding quadrature formulas of Gaussian type concerning the even weight function ωλ(t; x) = exp(−xt2)(1 − t2)λ−1/2 on (−1, 1), with parameters λ > −1/2 and x > 0, are considered. For λ = 1/2 these quadrature rules reduce to the so-called Gauss-Rys quadrature formulas, which were investigated earlier by several authors, e.g., Dupuis at al 1976 and 1983; Sagar 1992; Schwenke 2014; Shizgal 2015; King 2016; Milovanović 2018, etc. In this generalized case, the method of modified moments is used, as well as a transformation of quadratures on (−1, 1) with N nodes to ones on (0, 1) with only (N + 1)/2 nodes. Such an approach provides a stable and very efficient numerical construction.