Asymptotics of Discrete Chebyshev Polynomials

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, $N\to \mathrm{\infty }$ and $n\mathrm{/}N\to b$, where $b\in (0,1)$ and $a\in (-\mathrm{\infty },\mathrm{\infty })$; see [Studies in Appl. Math. $\mathbf{\text{128}}$ (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 $b\rightarrow 1$isrelativelysimple(sinceitisverymuchlikethecasewhen$b$ is fixed), the case $b\to 0$ is quite complicated. The discussion of the latter case is divided into several subcases, depending on the quantities $n$, $x$ and $xN\mathrm{/}{n}^2$, and different special functions have been used as approximants, including Airy, Bessel and Kummer functions.