Uniform Asymptotic Expansions for the Discrete Chebyshev Polynomials

Abstract: The discrete Chebyshev polynomials $t_n(x,N)$ are orthogonal with respect to a distribution function, which is a step function with jumps one unit at the points $x=0,1,..., N-1$, N being a fixed positive integer. By using a double integral representation, we derive two 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 $a\in(-\infty,\infty)$. One expansion involves the confluent hypergeometric function and holds uniformly for $a\in[0,1/2]$, and the other involves the Gamma function and holds uniformly for $a\in(-\infty, 0)$. Both intervals of validity of these two expansions can be extended slightly to include a neighborhood of the origin. Asymptotic expansions for $a\geq1/2$ can be obtained via a symmetry relation of $t_{n}(aN,N+1)$ with respect to $a=1/2$. Asymptotic formulas for small and large zeros of $t_{n}(x,N+1)$ are also given.