Asymptotic expansion of discrete Chebyshev polynomials

Abstract

The discrete Chebyshev polynomials tn(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. In this thesis, by using a double integral representation, we derive asymptotic expansions for tn(aN,N + 1) in the double scaling limit, namely, N→∞and n/N→b, where b ∈ (0, 1) and a ∈ (-∞,∞). Our discussion is divided into two parts. In the first part, we obtain two expansions for tn(aN,N+1) when the parameter b is a fixed constant in the interval (0, 1). One expansion involves the confluent hypergeometric function and holds uniformly for a ∈ [0, 1/2 ], 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 neighbourhood of the origin. Asymptotic expansions for a ≥ 1/2 can be obtained via a symmetry relation of tn(aN,N +1) with respect to a = 1/2 . Asymptotic formulas for small and large zeros of tn(x,N + 1) are also given. In the second part, we continue to investigate the behaviour of these polynomials when the parameter b approaches the endpoints of the interval (0, 1). While the case b→1 is relatively simple (since 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.

Date of Award	15 Jul 2013
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Sue Cheun Roderick WONG (Supervisor) & Dan DAI (Co-supervisor)