Asymptotics of Discrete Chebyshev Polynomials

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.