Experimenting with Apery Limits and WZ pairs

TL;DR

This paper demonstrates the power of experimental mathematics through experiments with Apery limits and WZ pairs, proving a weaker form of a conjecture, generating new limits, and rediscovering cubic irrationalities with promising irrationality measures.

Contribution

It introduces new experimental methods and results in the study of Apery limits, WZ pairs, and irrationalities, including a proof of a weaker conjecture and the discovery of a new family of cubic irrationalities.

Findings

Proved a weaker form of a conjecture by Chamberland and Straub.

Generated numerous new Apery limits.

Rediscovered a family of cubic irrationalities with promising irrationality measures.

Abstract

This article, dedicated with admiration in memory of Jon and Peter Borwein, illustrates by example, the power of experimental mathematics, so dear to them both, by experimenting with so-called Apery limits and WZ pairs. In particular we prove a weaker form of an intriguing conjecture of Marc Chamberland and Armin Straub (in an article dedicated to Jon Borwein), and generate lots of new Apery limits. We also rediscovered an infinite family of cubic irrationalities, that suggested very good effective irrationality measures (lower than Liouville's generic 3), and that seemed to go down to the optimal 2. It turned out this follows from known deep results in number theory, and a postscript by Paul Voutier outlines the proof. Nevertheless we believe that further experiments with our Maple packages will lead to new and interesting results.