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Asymptotics of the associated Pollaczek polynomials


Authors: Min-Jie Luo and R. Wong
Journal: Proc. Amer. Math. Soc. 147 (2019), 2583-2597
MSC (2010): Primary 33C45, 41A60; Secondary 33C05
DOI: https://doi.org/10.1090/proc/14405
Published electronically: February 20, 2019
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Abstract: In this note, we investigate the large-$ n$ behavior of the associated Pollaczek polynomials $ P_{n}^{\lambda }\left (z;a,b,c\right )$. These polynomials involve four real parameters $ \lambda $, $ a$, $ b$, and $ c$, in addition to the complex variable $ z$. Asymptotic formulas are derived for these polynomials, when $ z$ lies in the complex plane bounded away from the interval of orthogonality $ \left (-1,1\right )$, as well as in the interior of the interval of orthogonality. In the process of studying the asymptotic behavior of these polynomials when $ z\in \mathbb{C}\setminus [-1,1]$, we found that the existing representations of $ P_{n}^{\lambda }\left (z;a,b,c\right )$ do not provide useful information about their large-$ n$ asymptotics. Here, we present a new representation in terms of the Gauss hypergeometric functions, from which the large-$ n$ asymptotics for $ z$ in $ \mathbb{C}\setminus [-1,1]$ can be readily obtained. The asymptotic approximation in the interior of the interval of orthogonality is obtained by using asymptotic theory for difference equations.


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Additional Information

Min-Jie Luo
Affiliation: Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Hong Kong, China
Email: mathwinnie@live.com

R. Wong
Affiliation: Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Hong Kong, China

DOI: https://doi.org/10.1090/proc/14405
Keywords: Asymptotics, difference equations, hypergeometric functions, Pollaczek polynomials.
Received by editor(s): August 7, 2018
Received by editor(s) in revised form: September 12, 2018, and September 19, 2018
Published electronically: February 20, 2019
Additional Notes: The first author is the corresponding author.
Communicated by: Mourad Ismail
Article copyright: © Copyright 2019 American Mathematical Society