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 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...

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Bibliographic Details
Published in:Journal of difference equations and applications 2023-01, Vol.29 (1), p.32-42
Main Authors: Dougherty-Bliss, Robert, Zeilberger, Doron
Format: Article
Language:English
Subjects:
Apéry limits
Binomial coefficients
convergence acceleration
diophantine approximation
Initial conditions
Linear difference equations with polynomial coefficients
Rational functions
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Summary: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.
ISSN:1023-6198
1563-5120
DOI:10.1080/10236198.2022.2159817