Dubious Identities: A Visit to the Borwein Zoo

Abstract

We contribute to the zoo of dubious identities established by J.M. and P.B. Borwein in their 1992 paper, “Strange Series and High Precision Fraud” with five new entries, each of a different variety than the last. Some of these identities are again a high precision fraud and picking out the true from the bogus can be a challenging task with many unexpected twists along the way.

This work is dedicated to the Borwein family, mathematicians extraordinaire with a propensity for the implausible.

MSC:

• 33F10,40
• 41A58
• 41A99

ACKNOWLEDGMENT

The authors wish to thank Margarite L. LaBorde and Tanay V. Wakhare for their interesting comments on an early version of this work, as well as Chance Sanford for sending a rare reference related to Gosper’s work on infinite products of matrices. The first author acknowledges support from the Department of Defense SMART scholarship program. The second author thanks Tucker and Nivens for their unconditional support.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the authors.

Additional information

Notes on contributors

Zachary P. Bradshaw

ZACHARY P. BRADSHAW (ORCID 0000-0002-3591-7594) earned his Ph.D. in mathematics from Tulane University in 2023. His research interests are centered in mathematical physics with a focus on integration, quantum information, and quantum computation. In 2022, Zachary was awarded the Department of Defense’s SMART scholarship for service award, and he is now a research scientist at the Naval Surface Warfare Center, Panama City Division fulfilling his service obligation.

Department of Mathematics, Tulane University, New Orleans LA 70118

[email protected]

Christophe Vignat

CHRISTOPHE VIGNAT (ORCID 0000-0001-8739-8549) earned his Ph.D. in physics from Université Paris-Sud 11, Orsay, now known as Université Paris Saclay. He is now Professor at the physics department of this same university and a member of the Laboratoire des Signaux et Systèmes at CentraleSupélec. In the past, he has benefited from multiple invitations to the Tulane University department of mathematics. His research interests are centered around experimental mathematics, special functions and symbolic computation.

Department of Physics, Université Paris Saclay, L.S.S, CentraleSupélec, Orsay, 91190, France

Department of Mathematics, Tulane University, New Orleans LA 70118

[email protected], [email protected]