New Representations for all Sporadic Apéry-Like Sequences, With Applications to Congruences

Abstract

Sporadic Ap\'ery-like sequences were discovered by Zagier, by Almkvist and Zudilin and by Cooper in their searches for integral solutions for certain families of second- and third-order differential equations. We find new representations, in terms of constant terms of powers of Laurent polynomials, for all the 15 sporadic sequences. The new representations in turn lead to binomial expressions for the sequences, which, as opposed to previous expressions, do not involve powers of 8 and powers of 3. We use these to establish the supercongruence $B_{np^k} \equiv B_{np^{k-1}} \bmod p^{2k}$ for all primes $p \ge 3$ and integers $n,k \ge 1$, where $B_n$ is a sequence discovered by Zagier and known as Sequence $\mathbf{B}$. Additionally, for 14 out of the 15 sequences, the Newton polytopes of the Laurent polynomials used in our representations contain the origin as their only interior integral point. This property allows us to prove that these 14 sporadic sequences satisfy a strong form of the Lucas congruences, extending work of Malik and Straub. Moreover, we obtain lower bounds on the $p$-adic valuation of these 14 sequences via recent work of Delaygue.