Apéry Polynomials and the multivariate Saddle Point Method

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}^{\infty}) in the whole complex plane as (n\rightarrow \infty). 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}^{\infty}) 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.

arXiv comment19 pages