Uniform asymptotic expansion of the Jacobi polynomials in a complex domain

R. WONG; Yu-Qiu ZHAO

doi:10.1142/9789811290862_0001

Uniform asymptotic expansion of the Jacobi polynomials in a complex domain

R. WONG^*
, Yu-Qiu ZHAO

^*Corresponding author for this work

Department of Mathematics

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

Abstract

An asymptotic formula is found that links the behaviour of the Jacobi polynomial P_n^(α,β) (z) in the interval of orthogonality [−1, 1] with that outside the interval. The two infinite series involved in this formula are shown to be exponentially improved asymptotic expansions. The method used in this paper can also be adopted in other cases of orthogonal polynomials such as Hermite and Laguerre. © 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	1513-1530
Number of pages	18
Volume	4
ISBN (Electronic)	9789811290862, 9789811290879
ISBN (Print)	9789811290855
DOIs	https://doi.org/10.1142/9789811290862_0001
Publication status	Published - 2026

Bibliographical note

This is a reprint of: Wong, R., & Zhao, Y.-Q. (2004). Uniform asymptotic expansion of the Jacobi polynomials in a complex domain. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 460(2049), 2569-2586. https://doi.org/10.1098/rspa.2004.1296

Funding

The research of R.W. was partly supported by grants from the Research Grant Council of Hong Kong. The research of Y.-Q.Z. was partly supported by the National Natural Science Foundation of China, GuangDong Natural Science Foundation and a fund from the Advanced Research Centre of ZhongShan University.

Research Keywords

complex domain
exponential improvement
Jacobi polynomials
uniform asymptotic expansion

RGC Funding Information

RGC-funded

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N2 - An asymptotic formula is found that links the behaviour of the Jacobi polynomial Pn(α,β) (z) in the interval of orthogonality [−1, 1] with that outside the interval. The two infinite series involved in this formula are shown to be exponentially improved asymptotic expansions. The method used in this paper can also be adopted in other cases of orthogonal polynomials such as Hermite and Laguerre. © 2026 by World Scientific Publishing Co. Pte. Ltd.

AB - An asymptotic formula is found that links the behaviour of the Jacobi polynomial Pn(α,β) (z) in the interval of orthogonality [−1, 1] with that outside the interval. The two infinite series involved in this formula are shown to be exponentially improved asymptotic expansions. The method used in this paper can also be adopted in other cases of orthogonal polynomials such as Hermite and Laguerre. © 2026 by World Scientific Publishing Co. Pte. Ltd.

KW - complex domain

KW - exponential improvement

KW - Jacobi polynomials

KW - uniform asymptotic expansion

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DO - 10.1142/9789811290862_0001

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EP - 1530

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 -

Uniform asymptotic expansion of the Jacobi polynomials in a complex domain

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Research Keywords

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Uniform asymptotic expansion of the Jacobi polynomials in a complex domain

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