Armin Straub — Monatshefte für Mathematik — Volume 203, 2024, Pages 883-898
Abstract
For each of the \(15\) known sporadic Apéry-like sequences, we prove congruences modulo \(p^2\) that are natural extensions of the Lucas congruences modulo \(p\). This extends a result of Gessel for the numbers used by Apéry in his proof of the irrationality of \(\zeta(3)\). Moreover, we show that each of these sequences satisfies two-term supercongruences modulo \(p^{2r}\). Using special constant term representations recently discovered by Gorodetsky, we prove these supercongruences in the two cases that remained previously open.Corrigendum
Theorems 1.2, 1.3, and 5.1 are true for \( 0 \le k < p \) so that \( k \) is a \( p \)-adic digit, as expected for Lucas congruences. The published version, unfortunately, displays the incorrect inequality \( 0 \le k < n \) instead.
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BibTeX
@article{gessel-lucas-2024,
author = {Armin Straub},
title = {Gessel--Lucas congruences for sporadic sequences},
journal = {Monatshefte für Mathematik},
year = {2024},
volume = {203},
pages = {883--898},
doi = {10.1007/s00605-023-01894-3},
}