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

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∞xνe−x/2Jν(μx)Ln2ν(x)dx=2νΓ(ν+12)1πμ(sinθ)ν+12Cnν+12(cosθ), where μμ and νν are real parameters such that μ≥0μ≥0 and ν>−12, cosθ=μ2−1/4μ2+1/4, and Cnλ(x) is a Gegenbauer (ultraspherical) polynomial.