Exploring general Apéry limits via the Zudilin–Straub t-transform

Abstract

Inspired by a recent beautiful construction of Armin Straub and Wadim Zudilin, that ‘tweaked’ the sum of the sth 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.

Keywords:

• Linear difference equations with polynomial coefficients
• convergence acceleration
• diophantine approximation
• Apéry limits

Acknowledgments

Many thanks are due to Wadim Zudilin and the two referees, whose insightful and useful remarks considerably improved the exposition.

Disclosure statement

No potential conflict of interest was reported by the author(s).