Apéry Limits: Experiments and Proofs

Abstract

An important component of Apéry’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 Apéry limits and highlight connections to continued fractions as well as the famous theorems of Poincaré 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.

Keywords:

• MSC: Primary 11B37
• Secondary 11B83
• 11A55

Acknowledgments

We are grateful to Alan Sokal for improving the exposition by kindly sharing lots of careful suggestions and comments. We also thank Wadim Zudilin for helpful comments, including his suggestion at the end of Section 7, and references.

Additional information

Notes on contributors

Marc Chamberland

MARC CHAMBERLAND is the Myra Steele Professor of Mathematics at Grinnell College. He has published in various research areas, including differential equations, number theory, classical analysis, and experimental mathematics. His embrace of experimental mathematics twenty years ago is clearly traced to Jon Borwein. Fondly remembering Jon’s enthusiasm, love of new ideas, and infatuation with the number π, Marc is currently writing a book about that same magical constant.

Department of Mathematics and Statistics, Grinnell College, Grinnell, IA 50112, USA

[email protected]

Armin Straub

ARMIN STRAUB received his Ph.D. in mathematics from Tulane University in 2012 under the direction of Victor Moll and co-advised by Jon Borwein. After postdoctoral positions at the University of Illinois at Urbana-Champaign and the Max-Planck-Institut für Mathematik, Armin joined the University of South Alabama in 2015. He is forever grateful for the privilege of working with Jon (resulting in 13 joint publications), who was so generous in sharing ideas, resources, and advice. Armin misses Jon, his infectious enthusiasm, and his unique quick wit.

Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA

[email protected]