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arXiv · 2301.12248

Gessel-Lucas congruences for sporadic sequences

Armin Straub
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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.

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Armin Straub. 2023-01-28. Gessel-Lucas congruences for sporadic sequences. https://arxiv.org/abs/2301.12248

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