Asymptotic expansion of the Tricomi-Carlitz polynomials and their zeros

K. F. Lee; R. Wong

doi:10.1142/9789811290862_0011

Asymptotic expansion of the Tricomi-Carlitz polynomials and their zeros

K. F. Lee
, 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

The Tricomi–Carlitz polynomials [Formula Presented] are non-classical discrete orthogonal polynomials on the real line with respect to the step function whose jumps are[Formula Presented].In this paper, we derive an asymptotic expansion for [Formula Presented] as n → ∞, valid uniformly for bounded real t, where ν = n + 2α − 1/2 and α is a positive parameter. The validity for bounded t can be extended to unbounded t by using a sequence of rational functions introduced by Olde Daalhuis and Temme. The expansion involves the Airy functions and their derivatives. Error bounds are given for one-term and two-term approximations. Asymptotic formulas are also presented for the zeros of [Formula Presented]. © 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	1726-1748
Number of pages	23
Volume	4
ISBN (Electronic)	978-981-12-9087-9 , 978-981-12-9086-2
ISBN (Print)	978-981-12-9085-5
DOIs	https://doi.org/10.1142/9789811290862_0011
Publication status	Published - 2026

Bibliographical note

This is a reprint of: LEE, K. F., & Wong, R. (2014). Asymptotic expansion of the Tricomi-Carlitz polynomials and their zeros. Journal of Computational and Applied Mathematics, 265, 220-242. https://doi.org/10.1016/j.cam.2013.09.047

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abstract = "The Tricomi–Carlitz polynomials [Formula Presented] are non-classical discrete orthogonal polynomials on the real line with respect to the step function whose jumps are[Formula Presented].In this paper, we derive an asymptotic expansion for [Formula Presented] as n → ∞, valid uniformly for bounded real t, where ν = n + 2α − 1/2 and α is a positive parameter. The validity for bounded t can be extended to unbounded t by using a sequence of rational functions introduced by Olde Daalhuis and Temme. The expansion involves the Airy functions and their derivatives. Error bounds are given for one-term and two-term approximations. Asymptotic formulas are also presented for the zeros of [Formula Presented]. {\textcopyright} 2026 by World Scientific Publishing Co. Pte. Ltd.",

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N2 - The Tricomi–Carlitz polynomials [Formula Presented] are non-classical discrete orthogonal polynomials on the real line with respect to the step function whose jumps are[Formula Presented].In this paper, we derive an asymptotic expansion for [Formula Presented] as n → ∞, valid uniformly for bounded real t, where ν = n + 2α − 1/2 and α is a positive parameter. The validity for bounded t can be extended to unbounded t by using a sequence of rational functions introduced by Olde Daalhuis and Temme. The expansion involves the Airy functions and their derivatives. Error bounds are given for one-term and two-term approximations. Asymptotic formulas are also presented for the zeros of [Formula Presented]. © 2026 by World Scientific Publishing Co. Pte. Ltd.

AB - The Tricomi–Carlitz polynomials [Formula Presented] are non-classical discrete orthogonal polynomials on the real line with respect to the step function whose jumps are[Formula Presented].In this paper, we derive an asymptotic expansion for [Formula Presented] as n → ∞, valid uniformly for bounded real t, where ν = n + 2α − 1/2 and α is a positive parameter. The validity for bounded t can be extended to unbounded t by using a sequence of rational functions introduced by Olde Daalhuis and Temme. The expansion involves the Airy functions and their derivatives. Error bounds are given for one-term and two-term approximations. Asymptotic formulas are also presented for the zeros of [Formula Presented]. © 2026 by World Scientific Publishing Co. Pte. Ltd.

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UR - https://www.scopus.com/pages/publications/105042914233#tab=impact

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BT - The Selected Works of Roderick S C Wong

A2 - Dai, Dan

A2 - Zhou, Ding-Xuan

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ER -

Asymptotic expansion of the Tricomi-Carlitz polynomials and their zeros

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Asymptotic expansion of the Tricomi-Carlitz polynomials and their zeros

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