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Asymptotics of the Discrete Chebyshev Polynomials

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Abstract

The discrete Chebyshev polynomials math formula are orthogonal with respect to a distribution, which is a step function with jumps one unit at the points math formula, N being a fixed positive integer. By using a double integral representation, we have recently obtained asymptotic expansions for math formula in the double scaling limit, namely, math formula and math formula, where math formula and math formula; 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 math formula is relatively simple (because it is very much like the case when b is fixed), the case math formula is quite complicated. The discussion of the latter case is divided into several subcases, depending on the quantities n, x, and math formula, and different special functions have been used as approximants, including Airy, Bessel, and Kummer functions.

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