Hypergeometry inspired by irrationality questions

Tweaking the Beukers Integrals In Search of More Miraculous Irrationality Proofs A La Apery

There are only aleph-zero rational numbers, while there are 2 to the power aleph-zero real numbers. Hence the probability that a randomly chosen real number would be rational is 0. Yet proving rigorously that any specific, natural, real constant, is irrational is usually very hard, witness that there are still no proofs of the irrationality of the Euler-Mascheroni constant, the Catalan constant, or Zeta(5). Inspired by Frits Beukers' elegant rendition of Apery's seminal proofs of the irrationality of Zeta(2) and Zeta(3), and heavily using algorithmic proof theory, we systematically searched fo…

关于某些 $2$-adic zeta 值的无理性

On the irrationality of certain $2$-adic zeta values

本文证明了 Kubota-Leopoldt $2$-adic zeta 函数在特定区间内存在无理数值，通过构造显式有理函数并利用 Volkenborn 积分结合二项式系数操作实现。

Exploring General Apéry Limits via the Zudilin-Straub t-transform

Inspired by a recent beautiful construction of Armin Straub and Wadim Zudilin, that 'tweaked' the sum of the $s^{th}$ powers of the $n$-th row of Pascal's triangle, getting instead of sequences of numbers, sequences of rational functions, we do the same for general binomial coefficients sums, getting a practically unlimited supply of Apéry limits. While getting what we call "major Apéry miracles", proving irrationality of the associated constants (i.e. the so-called Apéry limits) is very rare, we do get, every time, at least a "minor Apéry miracle" where an explicit constant, defined as an (ex…