Asymptotics of the Discrete Chebyshev Polynomials

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Abstract

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

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