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Biorthogonal polynomials and numerical integration formulas for infinite intervals. (English) Zbl 1221.65081

The paper refers to a class of numerical quadrature formulas for the infinite-range integrals \(\int^\infty_0w(x)f(x)dx\), where \(w(x)=x^ae^{-x}\) or \(w(x)=x^a\int_1^\infty e^{-xt}t^{-p}dt\). Integrals \(\int^\infty_{\dot u}w(x)/(z-x)dx\) involving this \(w(x)\) arise in radiation theory. It is pointed out that these formulas are obtained by applying the Levin and Sidi transformations, effective convergence methods to the asymptotic expansions of \(\int^\infty_{\dot u}w(x)/(z-x) \,dx\) (see [A. Sidi, Math. Comput. 35, 851–874 (1980; Zbl 0441.41021)]). All that turns out to be interpolatory. The effectivness of the numerical quadrature formulas is shown in some tables.

MSC:

65D32 Numerical quadrature and cubature formulas
65D30 Numerical integration
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C47 Other special orthogonal polynomials and functions
40A25 Approximation to limiting values (summation of series, etc.)
41A20 Approximation by rational functions
41A55 Approximate quadratures
65B10 Numerical summation of series

Citations:

Zbl 0441.41021
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