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

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: View the MathML source∫0∞xνe−x/2Jν(μx)Ln2ν(x)dx=2νΓ(ν+12)1πμ(sinθ)ν+12Cnν+12(cosθ), Turn MathJax on where μμ and νν are real parameters such that μ≥0μ≥0 and View the MathML sourceν>−12, View the MathML sourcecosθ=μ2−1/4μ2+1/4, and View the MathML sourceCnλ(x) is a Gegenbauer (ultraspherical) polynomial.