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Biorthogonal polynomials and numerical quadrature formulas for some finite-range integrals with symmetric weight functions

Authors
Journal
Journal of Computational and Applied Mathematics
0377-0427
Publisher
Elsevier
Publication Date
Volume
272
Identifiers
DOI: 10.1016/j.cam.2014.05.013
Keywords
  • Biorthogonal Polynomials
  • Numerical Integration
  • Symmetric Weight Functions
  • Acceleration Of Convergence
  • Levin Transformation
  • Rational Approximation

Abstract

Abstract In this work, we derive a family of symmetric numerical quadrature formulas for finite-range integrals I[f]=∫−11w(x)f(x)dx, where w(x) is a symmetric weight function. In particular, we will treat the commonly occurring case of w(x)=(1−x2)α[log(1−x2)−1]p, p being a nonnegative integer. These formulas are derived by applying a modification of the Levin L transformation to some suitable asymptotic expansion of the function H(z)=∫−11w(x)/(z−x)dx as z→∞, and they turn out to be interpolatory. The abscissas of these formulas have some rather interesting properties: (i) they are the same for all α, (ii) they are real and in [−1,1], and (iii) they are related to the zeros of some known polynomials that are biorthogonal to certain powers of log(1−x2)−1. We provide tables and numerical examples that illustrate the effectiveness of our numerical quadrature formulas.

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