Construction and implementation of asymptotic expansions for Laguerre-type orthogonal polynomials

Laguerre and Laguerre-type polynomials are orthogonal polynomials on the interval [0,∞) with respect to a weight function of the form w(x) = xαe−Q(x), Q(x) = m ∑ k=0 qkx , α > −1, qm > 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 [28], based on a non-linear steepest descent analysis of an associated so-called Riemann–Hilbert problem. We will extend this work and show how to efficiently compute an arbitrary number of higherorder 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 paper. 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(− ∑m k=0 qkx ) on (−∞,∞), and to general non-polynomial functions Q(x) using contour integrals. The expansions may be used, e.g., to compute Gauss-Laguerre quadrature rules in a 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.