Orthogonal polynomials and generalized Gauss-Rys quadrature formulae

Authors

  • Gradimir Milovanovic Mathematical Institute Serbian Academy of Sciences and Arts Kneza Mihaila 36, 11000 Beograd, Serbia http://orcid.org/0000-0002-3255-8127
  • NevenaVasovic ́ UniversityofKragujevac,FacultyofHotelManagementandTourism,36210VrnjaˇckaBanja,Serbia

DOI:

https://doi.org/10.48129/kjs.v49i1.10665

Keywords:

Quadrature rule, orthogonal polynomials, recurrence relation, nodes, weights

Abstract

Orthogonal polynomials and the corresponding quadrature formulas of Gaussian type with respect to the even weight function
$\omega^{\lambda}(t;x)=\exp(-x t^2)(1-t^2)^{\lambda-1/2}$ 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, 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.

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Published

02-12-2021