Biorthogonal Polynomials and Numerical Integration Formulas for Infinite Intervals

In this work, we consider a class of numerical quadrature formulas for the infiniterange integrals R∞ 0 w(x)f(x) dx, where w(x) = xαe−x and w(x) = xEp(x), Ep(x) being the Exponential Integral. These formulas are obtained by applying the Levin L and Sidi S transformations, two effective convergence acceleration methods, to the asymptotic expansions of R∞ 0 w(x)/(z − x) dx as z → ∞, and they turn out to be interpolatory. In addition, their abscissas turn out to have some interesting properties: For example, if xni, i = 1, . . . , n, are the abscissas of the appropriate n-point formula, then the polynomial Qn i=1(z − xni) is orthogonal to some set of n real exponential functions, e−σnkx, k = 1, . . . , n, where σ−1 nk are the zeros of some known polynomials. We provide some tables and numerical examples that show the effectiveness of our numerical quadrature formulas. c © 2007 European Society of Computational Methods in Sciences and Engineering