Error bounds for the asymptotic expansions of the Hermite polynomials

TL;DR

This paper derives explicit, easy-to-compute error bounds for the asymptotic expansions of Hermite polynomials across different regions, using a novel branch cut technique and recursive coefficient calculations.

Contribution

It introduces a new branch cut method for error estimation and provides recursive formulas for expansion coefficients in Hermite polynomial asymptotics.

Findings

Explicit error bounds expressed with elementary functions

Applicable to outer, oscillatory, and turning point regions

Recursive procedures for coefficient computation

Abstract

In this paper, we present explicit and computable error bounds for the asymptotic expansions of the Hermite polynomials with Plancherel--Rotach scale. Three cases, depending on whether the scaled variable lies in the outer or oscillatory interval, or it is the turning point, are considered separately. We introduce the "branch cut" technique to express the error terms as integrals on the contour taken as the one-sided limit of curves approaching the branch cut. This new technique enables us to derive simple error bounds in terms of elementary functions. We also provide recursive procedures for the computation of the coefficients appearing in the asymptotic expansions.