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Gessel-Lucas congruences for sporadic sequences

Gessel-Lucas congruences for sporadic sequences
Armin Straub — Monatshefte für Mathematik — 2023

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.

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BibTeX

@article{gessel-lucas-2023,
    author = {Armin Straub},
    title = {Gessel--Lucas congruences for sporadic sequences},
    journal = {Monatshefte für Mathematik},
    year = {2023},
    doi = {10.1007/s00605-023-01894-3},
}