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 Apéry-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 $B_{n{p}^k}\equiv B_{n{p}^{k-1}}\mathrm{mod}{p}^{2k}$ for all primes $p\ge 3$ and integers $n,k\ge 1$, where B_n is a sequence discovered by Zagier, known as Sequence 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.

Keywords:

• Constant term
• Apéry numbers
• sporadic sequences

2010 AMS Subject Classification:

• 05A15
• 05A19
• 11A07

Acknowledgments

In addition to useful comments, corrections and suggestions, I would like to thank Armin Straub for crucial computational help in the proof of Proposition 3.3 for $(\eta )$, and Wadin Zudilin for letting me know of the survey [58]. I am grateful for the referee’s helpful comments regarding the exposition.

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Notes

Notes

1 An example of a binomial sum is (1.1). The interested reader may find a rigorous definition of “binomial sums” in [11].

2 Version 4.1, available from https://specfun.inria.fr/chyzak/mgfun.html.

3 Version 1.7.3, available from https://www3.risc.jku.at/research/combinat/software/ergosum/installation.html.

Additional information

Funding

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 851318).