Uniform Asymptotic Expansions for the Discrete Chebyshev Polynomials

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 → ∞ and n/N → b, where b ∈ (0, 1) and a ∈ (−∞, ∞). One expansion involves the confluent hypergeometric function and holds uniformly for , and the other involves the Gamma function and holds uniformly for a ∈(−∞, 0). Both intervals of validity of these two expansions can be extended slightly to include a neighborhood of the origin. Asymptotic expansions for can be obtained via a symmetry relation of t_n(aN, N + 1) with respect to . Asymptotic formulas for small and large zeros of t_n(x, N + 1) are also given.