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Apéry limits: Experiments and proofs

Apéry limits: Experiments and proofs
Marc Chamberland, Armin Straub — American Mathematical Monthly — Special issue in memory of Jonathan Borwein — Volume 128, Number 9, 2021, Pages 811-824

Abstract

An important component of Apéry's proof that $$\zeta (3)$$ is irrational involves representing $$\zeta (3)$$ as the limit of the quotient of two rational solutions to a three-term recurrence. We present various approaches to such Apéry limits and highlight connections to continued fractions as well as the famous theorems of Poincaré and Perron on difference equations. In the spirit of Jon Borwein, we advertise an experimental-mathematics approach by first exploring in detail a simple but instructive motivating example. We conclude with various open problems.

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BibTeX

@article{aperylimits-2021,
author = {Marc Chamberland and Armin Straub},
title = {Ap\'ery limits: Experiments and proofs},
journal = {American Mathematical Monthly},
year = {2021},
volume = {128},
number = {9},
pages = {811--824},
doi = {10.1080/00029890.2021.1962153},
}