Exploring General Ap\'ery Limits via the Zudilin-Straub t-transform

TL;DR

This paper extends the construction of Apéry limits using the Zudilin-Straub t-transform to a broader class of binomial coefficient sums, generating numerous new constants and providing methods for their rapid computation.

Contribution

It introduces a generalized approach to generating Apéry limits from binomial sums, expanding the scope of known constants and their computational techniques.

Findings

Generated a large family of new Apéry limits.

Provided explicit sequences for fast approximation of these constants.

Demonstrated the method's ability to produce 'minor Apéry miracles'.

Abstract

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 Ap\'ery limits. While getting what we call "major Ap\'ery miracles", proving irrationality of the associated constants (i.e. the so-called Ap\'ery limits) is very rare, we do get, every time, at least a "minor Ap\'ery miracle" where an explicit constant, defined as an (extremely slowly-converging) limit of some explicit sequence, is expressed as an Ap\'ery limit of some recurrence, with some initial conditions, thus enabling a very fast computation of that constant, with exponentially decaying error.