Apéry Polynomials and the multivariate Saddle Point Method

2013/07/01 by Thorsten Neuschel, Neuschel, Thorsten · 1 citation
Mathematics · #30E15 (Primary) 41A60 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:30E15 #msc:41A60

paper · pdf · doi:10.48550/arxiv.1307.0341

19 pages

arxiv created 2013/07/01 · arxiv updated 2013/07/02

Abstract

The Apéry polynomials and in particular their asymptotic behavior play an essential role in the understanding of the irrationality of ζ(3). In this paper, we present a method to study the asymptotic behavior of the sequence of the Apéry polynomials ((B_n)_n=1^∞) in the whole complex plane as (n→ ∞). The proofs are based on a multivariate version of the complex saddle point method. Moreover, the asymptotic zero distributions for the polynomials ((B_n)_n=1^∞) and for some transformed Apéry polynomials are derived by means of the theory of logarithmic potentials with external fields, establishing a characterization as the unique solution of a weighted equilibrium problem. The method applied is a general one, so that the treatment can serve as a model for the study of objects related to the Apéry polynomials.

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