Evaluation of integrals involving orthogonal polynomials: Laguerre polynomial and Bessel function example

Abstract

Using the theory of orthogonal polynomials, their associated recursion relations and differential formulas we develop a new method for evaluating integrals that include orthogonal polynomials. The method is illustrated by obtaining the following integral result that involves the Bessel function and associated Laguerre polynomial: ∫ 0 ∞ x ν e − x / 2 J ν ( μ x ) L n 2 ν ( x ) d x = 2 ν Γ ( ν + 1 2 ) 1 π μ ( sin θ ) ν + 1 2 C n ν + 1 2 ( cos θ ) , where μ and ν are real parameters such that μ ≥ 0 and ν > − 1 2 , cos θ = μ 2 − 1 / 4 μ 2 + 1 / 4 , and C n λ ( x ) is a Gegenbauer (ultraspherical) polynomial.