$q$-正割数与广义$q$-欧拉数的高阶分圆同余

中文摘要 AI 辅助

设$\text{A}(2n)$表示集合$\text{{1,2,…,2n}}$的升降交替排列的集合，且$E_{2n}(q)=\text{∑}_{σ\in\text{A}(2n)}q^{\operatorname{inv}(σ)}$。Andrews和Foata证明了$E_{2n}(q)\equiv q^{2n(n-1)}\pmod{(1+q)^2}$，Liu最近得到了三次方细化结果：$E_{2n}(q)\equiv q^{2n(n-1)}-\binom n2(1+q)^2\pmod{(1+q)^3}$。利用$q$-正割数的互反生成函数、高斯系数在$q=-1$处的三阶展开、有限差分以及Newton插值，我们证明了四阶细化结果：$E_{2n}(q)\equiv q^{2n(n-1)}-\binom n2(1+q)^2+\binom n2(2n^2-2n-3)(1+q)^3\pmod{(1+q)^4}$。更一般地，该递推给出了对任意指定$K$计算模$(1+q)^K$展开的有效算法。随后我们将相同的局部展开策略应用于Sagan和Zhang提出的广义$q$-欧拉数$E_{pn\mid p}(q)$。对每个素数$p$，我们证明了模$[p]_q^3$和$[p]_q^4$的一致同余式；其中四阶项由与${2p\brack p}_q$相关的中心$q$-Wolstenholme型商控制。由此，四阶正割同余式是通用高阶分圆方法的首个实例。

英文摘要

Let $\A(2n)$ denote the set of up--down alternating permutations of $\{1,2,\ldots,2n\}$, and let \[ E_{2n}(q)=\sum_{σ\in\A(2n)}q^{\operatorname{inv}(σ)}. \] Andrews and Foata proved that $E_{2n}(q)\equiv q^{2n(n-1)}\pmod{(1+q)^2}$, and Liu recently obtained the cubic refinement \[ E_{2n}(q)\equiv q^{2n(n-1)}-\binom n2(1+q)^2 \pmod{(1+q)^3}. \] Using the reciprocal generating function for the $q$-secant numbers, a third-order expansion of Gaussian coefficients at $q=-1$, finite differences, and Newton interpolation, we prove the fourth-order refinement \[ E_{2n}(q)\equiv q^{2n(n-1)}-\binom n2(1+q)^2 +\binom n2(2n^2-2n-3)(1+q)^3 \pmod{(1+q)^4}. \] More generally, the recurrence yields an effective procedure for computing the expansion modulo $(1+q)^K$ for any prescribed $K$. We then apply the same local-expansion strategy to the generalized $q$-Euler numbers $E_{pn\mid p}(q)$ of Sagan and Zhang. For every prime $p$, we prove uniform congruences modulo $[p]_q^3$ and $[p]_q^4$; the fourth-order term is governed by a central $q$-Wolstenholme-type quotient associated with ${2p\brack p}_q$. Thus the fourth-order secant congruence is the first case of a general higher-cyclotomic method.

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16 pages