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 $$\zeta (3)$$ . In this paper, we present a method to study the asymptotic behavior of the sequence of 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.