A general asymptotic expansion formula for integrals involving high-order orthogonal polynomials

A new method of evaluating overlap integrals involving orthogonal polynomials is proposed. The technique relies on purely algebraic manipulation of the associated recurrence coefficients. For a large class of polynomials and for sufficiently large orders, these coefficients can be written explicitly as Taylor series in terms of powers of ε = 1/n, where n is the polynomial order. Such decompositions are perfectly suited to the accurate numerical evaluation of integrals involving high-order polynomials. Examples include the numerical evaluation of integrals involving classical orthogonal polynomials such as Laguerre, Jacobi, and Gegenbauer. © 2009 Society for Industrial and Applied Mathematics.