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

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)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, pp being a nonnegative integer. These formulas are derived by applying a modification of the Levin LL transformation to some suitable asymptotic expansion of the function H(z)=∫−11w(x)/(z−x)dx as z→∞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 αα, (ii) they are real and in [−1,1][−1,1], and (iii) they are related to the zeros of some known polynomials that are biorthogonal to certain powers of log(1−x2)−1log(1−x2)−1. We provide tables and numerical examples that illustrate the effectiveness of our numerical quadrature formulas.