Uniform Asymptotic Expansions for the Discrete Chebyshev Polynomials

The discrete Chebyshev polynomialst _n(x,N) are orthogonal with respect to a distribution function, which is a step function with jumps one unit at the pointsx= 0, 1,...,N- 1,Nbeing a fixed positive integer. By using a double integral representation, we derive two asymptotic expansions fort _n(aN,N+ 1) in the double scaling limit, namely,N→∞ andn/N→b, whereb∈ (0, 1) anda∈ (-∞, ∞). One expansion involves the confluent hypergeometric function and holds uniformly for, and the other involves the Gamma function and holds uniformly fora∈ (-∞, 0). Both intervals of validity of these two expansions can be extended slightly to include a neighborhood of the origin. Asymptotic expansions forcan be obtained via a symmetry relation oft _n(aN,N+ 1) with respect to. Asymptotic formulas for small and large zeros oft _n(x,N+ 1) are also given. © 2011 by the Massachusetts Institute of Technology.