Construction and implementation of asymptotic expansions for Jacobi--type orthogonal polynomials
Alfredo Dea\~no, Daan Huybrechs, Peter Opsomer

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 goes to . These are defined on the interval with weight function , and a real, analytic and strictly positive function on . 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…
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