New representations for all sporadic Apéry-like sequences, with applications to congruences

Abstract

We find new representations, in terms of constant terms of powers of Laurent polynomials, for all the 15 sporadic Ap{é}ry-like sequences discovered by Zagier, Almkvist-Zudilin and Cooper. The new representations lead to binomial expressions for the sequences, which, as opposed to previous expressions, do not involve powers of 3 or 8. 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, known as Sequence $\mathbf{B}$. Additionally, for 14 of the 15 sequences, the Newton polytopes of the Laurent polynomials contain the origin as their only interior integral point. This property allows us to prove that these 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 sequences via recent work of Delaygue.