Error bounds for the asymptotic expansions of the Hermite polynomials

Wei Shi; Gergὄ Nemes; Xiang-Sheng Wang; Roderick Wong

doi:10.1142/9789811290862_0024

Error bounds for the asymptotic expansions of the Hermite polynomials

Wei Shi
, Gergὄ Nemes
, Xiang-Sheng Wang
, Roderick Wong

Department of Mathematics

Research output: Chapters, Conference Papers, Creative and Literary Works › Reprint in book › peer-review

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. © 2026 by World Scientific Publishing Co. Pte. Ltd.

Original language	English
Title of host publication	The Selected Works Of Roderick S. C. Wong
Editors	Dan DAI, Ding-Xuan ZHOU
Place of Publication	Singapore
Publisher	World Scientific Publishing Co. Pte Ltd
Pages	2022-2045
Number of pages	24
Volume	4
ISBN (Electronic)	9789811290862, 9789811290879
ISBN (Print)	9789811290855
DOIs	https://doi.org/10.1142/9789811290862_0024
Publication status	Published - 2026

Bibliographical note

This is a reprint of: Shi, W., Nemes, G., Wang, X.-S., & Wong, R. (2023). Error bounds for the asymptotic expansions of the Hermite polynomials. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 153(2), 417-440. https://doi.org/10.1017/prm.2021.90

Funding

We are very grateful to the anonymous referee for his/her careful reading and valuable suggestions which have helped to improve the presentation of this paper. The second author was supported by the JSPS Postdoctoral Research Fellowship No. P21020.

Research Keywords

asymptotic expansions
Error bounds
Hermite polynomials

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Error bounds for the asymptotic expansions of the Hermite polynomials. / Shi, Wei; Nemes, Gergὄ; Wang, Xiang-Sheng et al.
The Selected Works Of Roderick S. C. Wong. ed. / Dan DAI; Ding-Xuan ZHOU. Vol. 4 Singapore: World Scientific Publishing Co. Pte Ltd, 2026. p. 2022-2045.

Research output: Chapters, Conference Papers, Creative and Literary Works › Reprint in book › peer-review

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AU - Wong, Roderick

N1 - This is a reprint of: Shi, W., Nemes, G., Wang, X.-S., & Wong, R. (2023). Error bounds for the asymptotic expansions of the Hermite polynomials. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 153(2), 417-440. https://doi.org/10.1017/prm.2021.90

PY - 2026

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N2 - 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. © 2026 by World Scientific Publishing Co. Pte. Ltd.

AB - 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. © 2026 by World Scientific Publishing Co. Pte. Ltd.

KW - asymptotic expansions

KW - Error bounds

KW - Hermite polynomials

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SN - 9789811290855

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EP - 2045

BT - The Selected Works Of Roderick S. C. Wong

A2 - DAI, Dan

A2 - ZHOU, Ding-Xuan

PB - World Scientific Publishing Co. Pte Ltd

CY - Singapore

ER -

Error bounds for the asymptotic expansions of the Hermite polynomials

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Error bounds for the asymptotic expansions of the Hermite polynomials

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