Orthogonal polynomials and generalized Gauss-Rys quadrature formulae - Gradimir Milovanovic; Nevena Vasovic ́ (Kuwait Journal of Science)
Orthogonal polynomials and the corresponding quadrature formulas of Gaussian type with respect to the even weight function $ω^{λ}(t;x)=\exp(-x t^2)(1-t^2)^{λ-1/2}$ 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, Rys, King (1976 and 1983), Sagar (1992), Schwenke (2014), Shizgal (2015), King (2016), Milovanovi\'c (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.