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

Abstract

Let $E_n$ denote the number of alternating permutations of $\{1,\dots ,n\}$, equivalently characterized by ${\sum }_{n\ge 0}{E}_n{z}^n/n!=\sec z+\tan z$. For every $q\ge 1$, the sequence $(E_n\mathrm{mod}q{)}_{n\ge 0}$ 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 $d(p)=\mathrm{lcm}(p-1,4)$ together with \[ d(p^r)\mid p^{r-1}d(p), \qquad s(p^r)\le r, \] and Ramassamy conjectured that both bounds are attained for every $r\ge 1$. In this paper, we introduce an algebraic frequency expansion for the Euler zig-zag numbers using Hurwitz series over the coefficient ring $S_r=(\mathbb{Z}/p^r\mathbb{Z})[x]/(x^2+1)$. 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 \[ d(p^r)=p^{r-1}d(p) \qquad \text{for every odd prime $p$ and every $r\ge 1$}, \] 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 $s(p^r)\ne r$, and based on our findings we further conjecture \[ s(p^r)\ge r-2 \] for every odd prime $p$ and every $r\ge 2$.

17 Pages