Construction and implementation of asymptotic expansions for Laguerre-type orthogonal polynomials · arXivDesk
1612.07578Dec 22, 201628 pages, 5 figures, 29 references. The article mentioned is arXiv:math/0504604 and the implementation is available on http://nines.cs.kuleuven.be/software/LAGUERRE/
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)=∑k=0mqkxk,α>−1,qm>0
Nearby in the stack
. 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, 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 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 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(−∑k=0mqkx2k)
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.