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 $\sum_{n\ge0}E_nz^n/n!=\sec z+\tan z$. For every $q\ge1$, the sequence $(E_n\bmod q)_{n\ge0}$ 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 $d(p)=\operatorname{lcm}(p-1,4)$ 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 $r\ge1$. In this paper, we introduce an algebraic frequency expansion for the Euler numbers over $$S_r=(\mathbb Z/p^r\mathbb Z)[x]/(x<sup>2+1)$$. Using Hurwitz series, the Euler sequence is represented algebraically 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 $r\ge1$}, ] thereby establishing Ramassamy's period conjecture. We also disprove the preperiod conjecture by proving [ s(5^5)\le4<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 conjecture [ s(p^r)\ge r-2 ] for every odd prime $p$ and every $r\ge2$.