Gessel-Lucas congruences for sporadic sequences

Gessel-Lucas congruences for sporadic sequences
Armin Straub — Monatshefte für Mathematik — Volume 203, 2024, Pages 883-898


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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    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},