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

Ofir Gorodetsky

TL;DR

The paper proves that all 15 sporadic Apéry-like sequences admit constant-term representations as CTS(Λ^n) for Laurent polynomials Λ in 2–3 variables, with Λ expressible as a product of simple factors and with Newton polytopes having the origin as the only interior integral point for most sequences. This yields explicit binomial-sum representations for key sequences (notably B, F, and δ) and enables strong arithmetic results, including a supercongruence B_{np^k} ≡ B_{np^{k-1}} mod p^{2k} for p ≥ 3, as well as Lucas-type congruences and p-adic valuation bounds (via Delaygue) for almost all sequences. The approach unifies combinatorial, geometric, and p-adic techniques by exploiting diagonals, constant terms, and Newton polytope properties, and it extends the known congruence framework (Gauss, D3, Lucas) to a broader class of Apéry-like sequences. Additionally, the paper analyzes Legendrian and hypergeometric families and discusses future directions, including algorithmic recognition of CT representations and potential discovery of new Calabi–Yau-type relations. Overall, it provides a structural and arithmetic bridge between combinatorial representations and modular/Calabi–Yau phenomena for these sequences, with concrete new representations and congruence results that deepen understanding of their arithmetic nature.

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.

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

TL;DR

The paper proves that all 15 sporadic Apéry-like sequences admit constant-term representations as CTS(Λ^n) for Laurent polynomials Λ in 2–3 variables, with Λ expressible as a product of simple factors and with Newton polytopes having the origin as the only interior integral point for most sequences. This yields explicit binomial-sum representations for key sequences (notably B, F, and δ) and enables strong arithmetic results, including a supercongruence B_{np^k} ≡ B_{np^{k-1}} mod p^{2k} for p ≥ 3, as well as Lucas-type congruences and p-adic valuation bounds (via Delaygue) for almost all sequences. The approach unifies combinatorial, geometric, and p-adic techniques by exploiting diagonals, constant terms, and Newton polytope properties, and it extends the known congruence framework (Gauss, D3, Lucas) to a broader class of Apéry-like sequences. Additionally, the paper analyzes Legendrian and hypergeometric families and discusses future directions, including algorithmic recognition of CT representations and potential discovery of new Calabi–Yau-type relations. Overall, it provides a structural and arithmetic bridge between combinatorial representations and modular/Calabi–Yau phenomena for these sequences, with concrete new representations and congruence results that deepen understanding of their arithmetic nature.

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 for all primes and integers , where is a sequence discovered by Zagier, known as Sequence . 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 -adic valuation of these sequences via recent work of Delaygue.

Paper Structure

This paper contains 20 sections, 11 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Sporadic sequences
  3. Main results
  4. Gauss congruences
  5. Further results
  6. Lucas and D3 congruences
  7. D3 congruences for sporadic sequences
  8. p-adic valuation of sporadic sequences
  9. Constant term and binomial representations for sporadic sequences
  10. Review of diagonals, constant terms and binomial representations
  11. Known binomial sum representations
  12. Known diagonal representations
  13. New constant term representations
  14. Proof of Proposition \ref{['prop:formulas']}
  15. Proof of Theorem \ref{['thm:newton']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\{u_n \}_{n \ge 1}$ be one of the 15 sporadic sequences, and denote by $k \in \{2,3\}$ the order of the differential equation it satisfies. Then $u_n$ is a constant term sequence of a Laurent polynomial $\Lambda$ with integer coefficients in $k$ variables $x_1,\ldots,x_k$. In addition,

Theorems & Definitions (16)

  • Theorem 1.1
  • Proposition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Corollary 2.5
  • ...and 6 more