Asymptotics of Discrete Chebyshev Polynomials

TL;DR

This paper analyzes the asymptotic behavior of discrete Chebyshev polynomials as the parameters approach boundary values, revealing complex behaviors and employing special functions like Airy, Bessel, and Kummer for approximation.

Contribution

It extends previous asymptotic analysis of discrete Chebyshev polynomials to boundary cases where the parameter approaches zero or one, using diverse special functions for approximation.

Findings

Asymptotic behavior near boundary points is characterized.

Different special functions are used for different subcases.

Complex behaviors are uncovered as parameters approach endpoints.

Abstract

The discrete Chebyshev polynomials $t_n(x,N)$ are orthogonal with respect to a distribution, which is a step function with jumps one unit at the points $x=0,1,\cdots, N-1$, $N$ being a fixed positive integer. By using a double integral representation, we have recently obtained asymptotic expansions for $t_{n}(aN,N+1)$ in the double scaling limit, namely, $N\rightarrow\infty$ and $n/N\rightarrow b$, where $b\in (0,1)$ and $a\in(-\infty,\infty)$; see [Studies in Appl. Math. \textbf{128} (2012), 337-384]. In the present paper, we continue to investigate the behaviour of these polynomials when the parameter $b$ approaches the endpoints of the interval $(0,1)$. While the case $b\rightarrow 1$ is relatively simple (since it is very much like the case when $b$ is fixed), the case $b\rightarrow 0$ is quite complicated. The discussion of the latter case is divided into several subcases, depending on the quantities $n$, $x$ and $xN/n^2$, and different special functions have been used as approximants, including Airy, Bessel and Kummer functions.