Orthogonal polynomials and generalized Gauss-Rys quadrature formulae

Orthogonal polynomials and the corresponding quadrature formulas of Gaussian type concerning omega(lambda) (t; x) = exp (-xt(2)) (1 - t(2))(lambda-1/2) on (-1, 1), with parameters lambda > -1/2 and x >0, are considered. For lambda = 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; Milovanovic 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.