Construction And Implementation Of Asymptotic Expansions For Laguerre-Type Orthogonal Polynomials

Laguerre and Laguerre-type polynomials are orthogonal polynomials on the interval [0, infinity) with respect to a weight function of the formw(x) = x(alpha)e(-Q(x)), Q(x) = Sigma(m)(k=0)qk(xk), alpha > -1, q(m) > 0.The classical Laguerre polynomials correspond to Q(x) = x. The computation of higher order terms of the asymptotic expansions of these polynomials for large degree becomes quite complicated, and a full description seems to be lacking in literature. However, this information is implicitly available in the work of Vanlessen (2007, Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory. Constr. Approx., 25, 125-175), based on a nonlinear steepest descent analysis of an associated Riemann-Hilbert problem. We will extend this work and show how to efficiently compute an arbitrary number of higher order terms in the asymptotic expansions of Laguerre and Laguerre-type polynomials. This effort is similar to the case of Jacobi and Jacobi-type polynomials in a previous article. We supply an implementation with explicit expansions in four different regions of the complex plane. These expansions can also be extended to Hermite-type weights of the form exp(-Sigma(m)(k=0)qk(x2k)) on (-infinity,infinity) and to general nonpolynomial functions Q(x) using contour integrals. The expansions may be used, e.g., to compute Gauss-Laguerre quadrature rules with lower computational complexity than based on the recurrence relation and with improved accuracy for large degree. They are also of interest in random matrix theory.