Gessel-Lucas congruences for sporadic sequences

TL;DR

This paper proves new congruences modulo prime powers for all known sporadic Apéry-like sequences, extending classical Lucas congruences and establishing supercongruences, thus deepening understanding of their arithmetic properties.

Contribution

It extends Gessel's Lucas congruences to all 15 sporadic Apéry-like sequences and proves supercongruences using novel constant term representations.

Findings

Proved modulo p^2 congruences for all 15 sequences

Established two-term supercongruences modulo p^{2r}

Validated supercongruences in previously open cases

Abstract

For each of the $15$ known sporadic Ap\'ery-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 Ap\'ery 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.