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

By Robert Dougherty-Bliss and Doron Zeilberger

.pdf    .tex   

published in Journal of Difference Equations and Applications, v. 29(1) (2023), 34-42.

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 an arbitrary binomial coefficients sum, gettings a practically unlimited supply of Apéry limits. While getting what we call "major Apéry miracles", proving irrationality of the associated constant (i.e. the so-called Apéry limit) 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.

Written: May 2022.

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Sample Input and Output for Zudilin.txt

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