Ap\'ery Limits: Experiments and Proofs

TL;DR

This paper explores Apéry limits related to the irrationality proof of ζ(3), examining approaches involving recurrence relations, continued fractions, and difference equations, and emphasizes an experimental mathematics perspective with open problems.

Contribution

It introduces new methods and connections for analyzing Apéry limits, combining experimental mathematics with classical theorems on difference equations.

Findings

Connections between Apéry limits and continued fractions

Illustration of difference equations in Apéry's proof

Open problems in the study of Apéry limits

Abstract

An important component of Ap\'ery's proof that $\zeta (3)$ is irrational involves representing $\zeta (3)$ as the limit of the quotient of two rational solutions to a three-term recurrence. We present various approaches to such Ap\'ery limits and highlight connections to continued fractions as well as the famous theorems of Poincar\'e and Perron on difference equations. In the spirit of Jon Borwein, we advertise an experimental-mathematics approach by first exploring in detail a simple but instructive motivating example. We conclude with various open problems.