利用Zudilin-Straub t变换探索广义Apéry极限
Exploring General Apéry Limits via the Zudilin-Straub t-transform
摘要：
受Armin Straub和Wadim Zudilin最近的一个美丽的构造的启发，“调整”了Pascal三角形的第$n$行的$s^{th}$幂的和，得到的不是数字序列，而是有理函数序列，我们对一般的二项式系数和做了同样的事情，得到了几乎无限的Apéry极限。在得到我们所说的“重大Apéry奇迹”的同时，证明了相关常数的不合理性，（即所谓的Apéry极限）是非常罕见的，我们每次都至少得到一个“小Apéry奇迹”，其中一个明确的常数，定义为（极慢收敛）极限，表示为某种递推式的Apéry极限，并给出了某些初始条件，从而能够非常快速地计算该常数，误差呈指数衰减。
英文摘要：
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 (extremely slowly-converging) limit of some explicit sequence, is expressed as an Apéry limit of some recurrence, with some initial conditions, thus enabling a very fast computation of that constant, with exponentially decaying error.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
