Uniform Asymptotics of the Meixner Polynomials

Details Uniform Asymptotics of the Meixner Polynomials Uniform Asymptotics of the Meixner Polynomials

2008-11-17

arxiv.org/abs/0811.2624v3

Mathematics

Classical Analysis and ODEs

60 pages, 8 figures

Scientific paper

Using the steepest descent method of Deift-Zhou, we derive locally uniform asymptotic formulas for the Meixner polynomials. These include an asymptotic formula in a neighborhood of the origin, a result which as far as we are aware has not yet been obtained previously. This particular formula involves a special function, which is the uniformly bounded solution to a scalar Riemann-Hilbert problem, and which is asymptotically (as the polynomial degree $n$ tends to infinity) equal to the constant $"1"$ except at the origin. Numerical computation by using our formulas, and comparison with earlier results, are also given.

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