Metadata

bibkey: kimotowakayama2022apery
authors: Kimoto, Kazufumi; Wakayama, Masato
year: 2022
title: "Apéry-like numbers for non-commutative harmonic oscillators and automorphic integrals"
doi: 10.4171/aihpd/129
url: "https://arxiv.org/abs/1905.01775v2"
claim: "The paper defines Apéry-like numbers J_k(n) from the first anomaly of the spectral zeta function of the non-commutative harmonic oscillator, normalizes them to rational numbers satisfying a common three-term inhomogeneous recurrence, relates their generating functions to automorphic integrals, and conjectures congruences modulo prime powers among the normalized numbers (Conjecture 3.6), proving weaker versions."
strata_touched:
  - D5/S3/Arith/Congruence/KimotoWakayamaEvenCongruenceRefutation
license: citation-only
triage: anchor

Kimoto and Wakayama, Apéry-like numbers for non-commutative harmonic oscillators and automorphic integrals

K. Kimoto and M. Wakayama, Ann. Inst. Henri Poincaré D 10 (2023), 205–275 (published online 2022), DOI 10.4171/aihpd/129; arXiv:1905.01775v2. Quotations are from the arXiv v2 TeX source.

The normalized Apéry-like numbers $\overset{_{k} (n) are rational numbers satisfying, for n \geq 2, 4 n^{2} J}{J}_{k} (n) - (8 n^{2} - 8 n + 3) \overset{_{k} (n - 1) + 4 (n - 1)^{2} J}{J}_{k} (n - 2) = 4 \overset{_{k - 2} (n - 1), with J}{J}_{2} (n) = \sum_{j = 0}^{n} (- 1)^{j} (j - 1/2)^{2} (j n)$ , $\overset{_{k} (0) = 1 for k = 1, 2 and 0 otherwise, and J}{J}_{k} (1)$ equal to $\frac{2}{3}, \frac{3}{4}, 1, 1$ for $k = 1, 2, 3, 4$ and $0$ otherwise.

Conjecture 3.6 states, for positive integers $m, n, s$ such that $m p \geq s$ and $m p^{n - 1} \geq s$ , two congruences modulo $p^{n}$ ; the second reads

$\frac{p ^{2 s n} J _{2 s + 2} ( m p ^{n} )}{p ^{2 s} J _{2 s + 2} ( m p )} \equiv \frac{p ^{2 s (n - 1)} J _{2 s + 2} ( m p ^{n - 1} )}{p ^{2 s} J _{2 s + 2} ( m p )} (\neq \equiv 0) (mod p^{n}) .$

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DOI: https://doi.org/10.4171/aihpd/129
Source: https://arxiv.org/abs/1905.01775v2

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Kimoto and Wakayama, Apéry-like numbers for non-commutative harmonic oscillators and automorphic integrals

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