Asymptotics of Pseudo-Jacobi Polynomials with Varying Parameters

Z. Song; R. Wong

doi:10.1142/9789811290862_0017

Asymptotics of Pseudo-Jacobi Polynomials with Varying Parameters

Z. Song
, R. Wong^*

^*Corresponding author for this work

Department of Mathematics

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

Abstract

In this paper, we study the asymptotic behavior of the Pseudo-Jacobi polynomials Pn(z; a, b) as n → ∞ for z in the whole complex plane. These polynomials are also known as the Romanovski–Routh polynomials. They occur in quantum mechanics, quark physics, and random matrix theory. When the parameter a is fixed or a > −n, there is no real-line orthogonality. Here, we consider the case when the parameters a and b depend on n; more precisely, we assume a = −(An + A₀), A > 1 and b = Bn + B₀, where A, B, A₀, B₀ are real constants. Our main tool is the asymptotic method developed for differential equations with a large parameter. © 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
Publisher	World Scientific Publishing Co. Pte Ltd
Pages	1824-1862
Volume	4
ISBN (Electronic)	9789811290862, 9789811290879
ISBN (Print)	9789811290855
DOIs	https://doi.org/10.1142/9789811290862_0017
Publication status	Published - Feb 2026

Bibliographical note

This is a reprint of: Song, Z., & Wong, R. (2017). Asymptotics of Pseudo-Jacobi Polynomials with Varying Parameters. Studies in Applied Mathematics, 139(1), 179-217. https://doi.org/10.1111/sapm.12177

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N2 - In this paper, we study the asymptotic behavior of the Pseudo-Jacobi polynomials Pn(z; a, b) as n → ∞ for z in the whole complex plane. These polynomials are also known as the Romanovski–Routh polynomials. They occur in quantum mechanics, quark physics, and random matrix theory. When the parameter a is fixed or a > −n, there is no real-line orthogonality. Here, we consider the case when the parameters a and b depend on n; more precisely, we assume a = −(An + A0), A > 1 and b = Bn + B0, where A, B, A0, B0 are real constants. Our main tool is the asymptotic method developed for differential equations with a large parameter. © 2026 by World Scientific Publishing Co. Pte. Ltd.

AB - In this paper, we study the asymptotic behavior of the Pseudo-Jacobi polynomials Pn(z; a, b) as n → ∞ for z in the whole complex plane. These polynomials are also known as the Romanovski–Routh polynomials. They occur in quantum mechanics, quark physics, and random matrix theory. When the parameter a is fixed or a > −n, there is no real-line orthogonality. Here, we consider the case when the parameters a and b depend on n; more precisely, we assume a = −(An + A0), A > 1 and b = Bn + B0, where A, B, A0, B0 are real constants. Our main tool is the asymptotic method developed for differential equations with a large parameter. © 2026 by World Scientific Publishing Co. Pte. Ltd.

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ER -

Asymptotics of Pseudo-Jacobi Polynomials with Varying Parameters

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