Quadrature formulas for integrals transforms generated by orthogonal polynomials

TL;DR

This paper develops quadrature formulas for integral transforms linked to classical orthogonal polynomials, such as Hermite, Laguerre, and Jacobi, using recurrence relations and asymptotic analysis.

Contribution

It introduces a unified approach to derive quadrature formulas for various integral transforms generated by classical orthogonal polynomials.

Findings

Quadrature formulas for Poisson-related integral transforms are derived.

The formulas apply to Fourier, Bessel, and Appell transforms.

The approach leverages recurrence relations and asymptotic expressions.

Abstract

By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, the Christoffel-Darboux-type bilinear generating function and their asymptotic expressions, we obtain quadrature formulas for integral transforms generated by the classical orthogonal polynomials. These integral transforms, related to the so-called Poisson integrals, correspond to a modified Fourier Transform in the case of the Hermite polynomials, a Bessel Transform in the case of the Laguerre polynomials and to an Appell Transform in the case of the Jacobi polynomials.