AI 中文总结
本文通过算术分组和余切幂和公式，给出了 Wang--Sun 仿射错排和式的闭式表达式，并证明了其整值性、分母性质及符号交替规律。
AI 中文摘要
设 $n$ 为偶数，$\zeta$ 为 $n$ 次本原单位根，$\Delta_n$ 为仿射群 $\overrightarrow{GL}(\mathbb{Z}/n\mathbb{Z})$ 中错排的集合。我们计算 Wang--Sun 型和式 \\[ S_n=\sum_{\pi\in\Delta_n}\mathrm{sign}(\pi) \prod_{x\in\mathbb{Z}/n\mathbb{Z}} \frac{1+\zeta^{x-\pi(x)}}{1-\zeta^{x-\pi(x)}}. \\] 主要步骤是算术分组：对于仿射映射 $\pi(x)=vx+u$，该乘积仅依赖于 $\sigma=\gcd(1-v,n)$ 以及 $u$ 模 $\sigma$ 的剩余类。对平移符号求和后，整个和式变为对偶数除数的求和，并带有明确的局部重数。仅剩的分析模块是交替的偶数余切幂和；Liu 和 Xin 的一个近期公式随后给出了在 Bernoulli 多项式和普适系数下的闭式表达式。此外，利用 Cvijović 和 Klinowski 的结果，我们可以仅用有理算术来求余切幂和。这为完整的仿射和式提供了一个一般公式，其初值在早期工作中仅通过定义计算。该公式还使我们能够证明该和式的一些一般性质：对于 $2$ 的幂次，其值为整数；对于素数 $p>3$，其分母为 $2p$；并且它具有交替符号。
英文摘要
Let $n$ be even, let $ζ$ be a primitive $n$-th root of unity, and let $Δ_n$ be the set of derangements in the affine group $\overrightarrow{GL}(\mathbb{Z}/n\mathbb{Z})$. We evaluate the Wang--Sun type sum \[ S_n=\sum_{π\inΔ_n}\mathrm{sign}(π) \prod_{x\in\mathbb{Z}/n\mathbb{Z}} \frac{1+ζ^{x-π(x)}}{1-ζ^{x-π(x)}}. \] The main step is an arithmetic grouping: for an affine map $π(x)=vx+u$, the product depends only on $σ=\gcd(1-v,n)$ and on the residue class of $u$ modulo $σ$. After summing the translation signs, the full sum becomes a sum over even divisors with explicit local multiplicities. The only remaining analytic blocks are alternating even cotangent power sums; a recent formula of Liu and Xin then gives a closed expression in Bernoulli polynomials and universal coefficients. Moreover, using results by Cvijović and Klinowski we can evaluate the cotangent power sums using exclusively rational arithmetic. This supplies a general formula for the full affine sum whose initial values were computed using the definition only in an earlier work. The formula also lets us prove some general patterns of the sum: it has integral value for powers of $2$, prime denominator for $2p$ for primes $p>3$ and it has alternating signs.
