Asymptotics of the 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, . . . , 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 → ∞ 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. © 2026 by World Scientific Publishing Co. Pte. Ltd.