Asymptotics of the Discrete Chebyshev Polynomials

The discrete Chebyshev polynomials tn(x,N) are orthogonal with respect to a distribution, 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 have recently obtained asymptotic expansions for tn(aN,N+1) in the double scaling limit, namely, N→∞ and n/N→b, where b∈(0,1) and a∈(-∞,∞); see [8]. In this paper, we continue to investigate the behavior of these polynomials when the parameter b approaches the endpoints of the interval (0, 1). While the case b→1 is relatively simple (because it is very much like the case when b is fixed), the case b→0 is quite complicated. The discussion of the latter case is divided into several subcases, depending on the quantities n, x, and xN/n2, and different special functions have been used as approximants, including Airy, Bessel, and Kummer functions. © 2013 by the Massachusetts Institute of Technology.