Modular periodicity of the Euler up/down numbers at odd prime powers

Abstract

Let E_n denote the number of alternating permutations of {1,…,n}, equivalently characterized by ∑_n≥0E_nzⁿ/n!=sec z+tan z. For every q≥1, the sequence (E_nbmod q)_n≥0 is eventually periodic; let d(q) and s(q) denote its minimal eventual period and preperiod. For every odd prime p, Knuth and Buckholtz proved d(p)=lcm(p−1,4) together with d(p^r)| p^r−1d(p), s(p^r)≤ r, and Ramassamy conjectured that both bounds are attained for every r≥1. In this paper, we introduce an algebraic frequency expansion for the Euler zig-zag numbers using Hurwitz series over the coefficient ring S_r=(ℤ/p^rℤ)[x]/(x²+1). More precisely, the corresponding Hurwitz series is represented as a finite combination of formal exponential modes, in a manner reminiscent of Fourier analysis. Using this expansion, we prove d(p^r)=p^r−1d(p) for every odd prime $p$ and every $r≥1$, thereby establishing Ramassamy's period conjecture. We also disprove the preperiod conjecture by proving s(5⁵)=4<5. Finally, we prove that 5⁵ is the smallest odd prime power for which s(p^r)≠ r, and based on our findings we further conjecture s(p^r)≥ r−2 for every odd prime p and every r≥2.

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BibTeX RIS

BibTeX

@misc{gulec2026modular,
  title = {{Modular periodicity of the Euler up/down numbers at odd prime powers}},
  author = {Berke Güleç},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.27058},
  eprint = {2608.27058},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27058},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Modular periodicity of the Euler up/down numbers at odd prime powers
AU  - Berke Güleç
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2608.27058
UR  - https://arxiv.org/abs/2608.27058
AB  - Let E_n denote the number of alternating permutations of {1,…,n}, equivalently characterized by ∑_n≥0E_nz^n/n!=sec z+tan z. For every q≥1, the sequence (E_nbmod q)_n≥0 is eventually periodic; let d(q) and s(q) denote its minimal eventual period and preperiod. For every odd prime p, Knuth and Buckholtz proved d(p)=lcm(p−1,4) together with d(p^r)| p^r−1d(p), s(p^r)≤ r, and Ramassamy conjectured that both bounds are attained for every r≥1. In this paper, we introduce an algebraic frequency expansion for the Euler zig-zag numbers using Hurwitz series over the coefficient ring S_r=(ℤ/p^rℤ)[x]/(x^2+1). More precisely, the corresponding Hurwitz series is represented as a finite combination of formal exponential modes, in a manner reminiscent of Fourier analysis. Using this expansion, we prove d(p^r)=p^r−1d(p) for every odd prime $p$ and every $r≥1$, thereby establishing Ramassamy's period conjecture. We also disprove the preperiod conjecture by proving s(5^5)=4<5. Finally, we prove that 5^5 is the smallest odd prime power for which s(p^r)≠ r, and based on our findings we further conjecture s(p^r)≥ r−2 for every odd prime p and every r≥2.
ER  -

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