New method for evaluating integrals involving orthogonal polynomials: Laguerre polynomial and Bessel function example

TL;DR

This paper introduces a novel method leveraging orthogonal polynomial theory to evaluate complex integrals, demonstrated through a closed-form involving Bessel functions and Laguerre polynomials expressed via Gegenbauer polynomials.

Contribution

It develops a new approach using recursion and differential relations of orthogonal polynomials to compute integrals involving special functions.

Findings

Derived a closed-form integral involving Bessel and Laguerre functions

Expressed the integral result in terms of Gegenbauer polynomials

Validated the method with a specific integral example

Abstract

Using the theory of orthogonal polynomials, their associated recursion relations and differential formulas we develop a method for evaluating new integrals. The method is illustrated by obtaining a closed-form expression for the value of an integral that involves the Bessel function and associated Laguerre polynomial. The result, which is given by Eq. (19) in the text, is written in terms of the Gegenbauer (ultra-spherical) polynomial.