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: integral(infinity)(0) x(v)e(-x/2) J(v)(mu x)L-n(2v)(x)dx = 2(v)Gamma (v + 1/2) 1/root pi eta (sin theta)(v+1/2) C-n(v)+(1/2) (cos theta), where mu and v are real parameters such that mu >= 0 and v > -1/2, cos theta mu 2-1/4/mu 2+1/4, and C-n(lambda)(x) is a Gegenbauer (ultraspherical) polynomial. (c) 2006 Elsevier Ltd. All rights reserved.