Ap\'ery Polynomials and the multivariate Saddle Point Method

TL;DR

This paper develops a multivariate saddle point method to analyze the asymptotic behavior and zero distributions of Apéry polynomials across the complex plane, advancing understanding of their properties related to ta(3).

Contribution

It introduces a general multivariate saddle point approach for studying Apéry polynomials' asymptotics and zero distributions, providing a new analytical framework.

Findings

Asymptotic behavior of Apéry polynomials is characterized in the complex plane.

Zero distributions are described via weighted equilibrium measures.

The method serves as a model for related polynomial asymptotic analyses.

Abstract

The Ap\'ery polynomials and in particular their asymptotic behavior play an essential role in the understanding of the irrationality of \zeta(3). In this paper, we present a method to study the asymptotic behavior of the sequence of the Ap\'ery 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 Ap\'ery 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 Ap\'ery polynomials.