Asymptotics of the associated Pollaczek polynomials

Luo, Min-Jie; Wong, R.

doi:10.1090/proc/14405

Asymptotics of the associated Pollaczek polynomials
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by Min-Jie Luo and R. Wong

Proc. Amer. Math. Soc. 147 (2019), 2583-2597

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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