Gaussian type quadrature rules related to the oscillatory modification of the generalized Laguerre weight functions.

In this paper we consider weighted integrals with respect to a modification of the generalized Laguerre weight functions w α (x) = x α e − x , α > − 1 , on (0 , + ∞) by the oscillatory factor exp (i ζ x) , where ζ > 0. For calculation of such integrals for analytic real-valued functions in D = { z ∈ ℂ ∣ Re z ≥ 0 , Im z ≥ 0 } we construct polynomials orthogonal with respect to the linear functional L : P → ℂ , given by L [ p ] = ∫ 0 + ∞ p (x) w α (x) exp (i ζ x) d x , where P is a linear space of all algebraic polynomials, as well as the corresponding quadrature rules of Gaussian type. The existence of such orthogonal polynomials and the corresponding three-term recurrence relations are proved. A few numerical examples are presented, including computation of the Cauchy principal value integrals. [ABSTRACT FROM AUTHOR]