The birthday boy problem

TL;DR

This paper proposes a strategy to explicitly identify irrational numbers defined by multiple integrals, enhancing the work of previous methods that proved their irrationality but did not specify the numbers.

Contribution

It introduces a method to explicitly identify and evaluate conjectured irrational numbers from integral representations, complementing existing irrationality proofs.

Findings

Successfully identified several conjectured numbers

Demonstrated the effectiveness of the proposed strategy

Provided explicit evaluations for multiple integral-based numbers

Abstract

In their recent preprint arXiv:2101.08308, Robert Dougherty-Bliss, Christoph Koutschan and Doron Zeilberger come up with a powerful strategy to prove the irrationality, in a quantitative form, of some numbers that are given as multiple integrals or quotients of such. What is really missing there, for many examples given, is an explicit identification of those irrational numbers; the authors comment on this task, "The output file [...] contains many such conjectured evaluations, (very possibly many of them are equivalent via a hypergeometric transformation rule) and we challenge [...], the birthday boy, or anyone else, to prove them." Without an identification, the numbers are hardly appealing to human (number theorists). The goal of this note is to outline a strategy to do the job and illustrate it on several promising entries discussed in the preprint above.