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

TL;DR

This paper develops simplified methods for asymptotic expansions of Jacobi-type orthogonal polynomials, enabling efficient and accurate computation of these polynomials and related quadrature rules for large degrees.

Contribution

It introduces a streamlined approach to higher-order asymptotic expansions, improving computational efficiency and accuracy for Jacobi-type orthogonal polynomials.

Findings

Asymptotic expansions are efficiently constructed symbolically and numerically.

Computations of polynomials and quadrature rules are faster, with complexity independent of degree.

The method outperforms traditional recurrence-based approaches in accuracy and speed.

Abstract

We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree $n$ goes to $\infty$. These are defined on the interval $[-1,1]$ with weight function $w(x)=(1-x)^{\alpha}(1+x)^{\beta}h(x)$, $\alpha,\beta>-1$ and $h(x)$ a real, analytic and strictly positive function on $[-1,1]$. This information is available in the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen, where the authors use the Riemann--Hilbert formulation and the Deift--Zhou non-linear steepest descent method. We show that computing higher-order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available. The main advantage of these expansions is that they lead to increasing accuracy for increasing degree of the polynomials, at a computational cost that is actually independent of the degree. In contrast, the typical use of the recurrence relation for orthogonal polynomials in computations leads to a cost that is at least linear in the degree. Furthermore, the expansions may be used to compute Gaussian quadrature rules in $\mathcal{O}(n)$ operations, rather than $\mathcal{O}(n^2)$ based on the recurrence relation.