Uniform Asymptotic Expansions for the Discrete Chebyshev Polynomials

TL;DR

This paper derives uniform asymptotic expansions for discrete Chebyshev polynomials in a double scaling limit, using integral representations to obtain formulas involving hypergeometric and Gamma functions, applicable across different parameter ranges.

Contribution

It introduces new uniform asymptotic expansions for discrete Chebyshev polynomials in the double scaling limit, extending their validity and providing formulas involving special functions.

Findings

Asymptotic expansions involving hypergeometric and Gamma functions

Uniform validity across specified parameter ranges

Asymptotic formulas for zeros of the polynomials

Abstract

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, $N\rightarrow\infty$ and $n/N\rightarrow b$, where $b\in(0,1)$ and $a\in(-\infty,\infty)$. One expansion involves the confluent hypergeometric function and holds uniformly for $a\in[0,1/2]$, and the other involves the Gamma function and holds uniformly for $a\in(-\infty, 0)$. Both intervals of validity of these two expansions can be extended slightly to include a neighborhood of the origin. Asymptotic expansions for $a\geq1/2$ can be obtained via a symmetry relation of $t_{n}(aN,N+1)$ with respect to $a=1/2$. Asymptotic formulas for small and large zeros of $t_{n}(x,N+1)$ are also given.