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

Abstract

Laguerre and Laguerre-type polynomials are orthogonal polynomials on the interval [0, ∞) with respect to a weight function of the formThe 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 q k x 2k ) 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.The methodology of [28] is based on the non-linear steepest descent method by Deift and Zhou [9] for a 2 × 2 Riemann-Hilbert problem that is generically associated with orthogonal polynomials by Fokas, Its and Kitaev [10]. This is further detailed in § 2.1. The general strategy is to apply a sequence of transformations, such that the final matrix-valued function R(z) is asymptotically close to the identity matrix as n or z tends to ∞. The asymptotic result for Y (z), and subsequently for the polynomials, is obtained by inverting these transformations. The transformations involve a normalization of behaviour at infinity, the so-called 'opening of a lens' around the interval of orthogonality, and the introduction of local parametrices in disks around special points like endpoints, which are matched to global parametrices elsewhere in the complex plane. These transformations split C into different regions, where different formulas for the asymptotics are valid.In our case of Laguerre-type polynomials, one also first needs an n-dependent rescaling of the x axis using the so-called MRS numbers (defined further on in this paper). After this step the roots of the rescaled polynomials accumulate in a fixed and finite interval. We provide an algorithm to obtain an arbitrary number of terms in the expansions, where we set up series expansions using many convolutions that follow the chain of transformations and their inverses in this steepest descent method. While doing this, we keep computational efficiency in mind, as well as the use of the correct branch cuts in the complex plane.The strategy outlined above and in § 2.1 was also followed in our earlier article about asymptotic expansions...

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