Gessel-Lucas congruences for sporadic sequences

2023/01/28 by Straub, Armin
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2301.12248

Abstract

For each of the 15 known sporadic Apéry-like sequences, we prove congruences modulo p² 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 ζ(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.

Related

• Divisibility properties of sporadic Apéry-like numbers
• On the binomial transforms of Apéry-like sequences
• Congruences for sporadic sequences and modular forms for non-congruence\n subgroups
• Lucas congruences using modular forms
• Lucas congruences using modular forms