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Orthogonal polynomials and generalized Gauss-Rys quadrature formulae. (English) Zbl 1499.65090

Summary: 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., M. Dupuis, J. Rys and H. F. King [“Evaluation of molecular integrals over Gaussian basis functions”, J. Chem. Phys. 65, 111–116 (1976)], R. P. Sagar and V. H. Smith [“On the calculation of Rys polynomials and quadratures”, Int. J. Quant. Chem. 42, No. 4, 827–836 (1992)], D. W. Schwenke [Comput. Phys. Commun. 185, No. 3, 762–763 (2014; Zbl 1360.81024)], B. D. Shizgal [“A novel Rys quadrature algorithm for use in the calculation of electron repuslion integrals”, Comput. Theor. Chem. 1074, 178–184 (2015)], H. F. King [“Strategies for evaluation of Rys roots and weights”, J. Phys. Chem. A 120, 9348–9351 (2016)], G. V. Milovanović [Bull., Cl. Sci. Math. Nat., Sci. Math. 43, 39–64 (2018; Zbl 1469.65068)], 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.

MSC:

65D32 Numerical quadrature and cubature formulas
33C47 Other special orthogonal polynomials and functions
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