论文标题
涉及危险和统一根源的猜想的证明
Proof of a conjecture involving derangements and roots of unity
论文作者
论文摘要
令$ n> 1 $为奇数。对于任何原始的$ n $ - th root $ζ$在复杂字段中的统一。通过EngenVector-eigenValue身份，我们表明$ \ sum_ {τ\ in D（n-1）} \ Mathrm {signrm {sign}（τ）（τ）\ prod_ {j = 1}^{n-1}^{n-1} {n-1} \ \ \\\\\\\\ \\ freac {1+e {1+e（ =（-1）^{\ frac {n-1} {2}}} \ frac {（（n-2）!!）这证实了Z.-W的先前猜想。太阳。此外，对于每个$δ= 0,1 $，我们确定$ \ det [x+m_ {jk}] _ {1 {1 \ le j，k \ le n} $的值j \ not = k，\\δ＆\ text {if} \ j = k。 \ end {case} $$
Let $n>1$ be an odd integer. For any primitive $n$-th root $ζ$ of unity in the complex field. Via the Engenvector-eigenvalue Identity, we show that $$\sum_{τ\in D(n-1)}\mathrm{sign}(τ)\prod_{j=1}^{n-1}\frac{1+ζ^{j-τ(j)}}{1-ζ^{j-τ(j)}} =(-1)^{\frac{n-1}{2}}\frac{((n-2)!!)^2}{n}, $$ where $D(n-1)$ is the set of all derangements of $1,\ldots,n-1$. This confirms a previous conjecture of Z.-W. Sun. Moreover, for each $δ=0,1$ we determine the value of $\det[x+m_{jk}]_{1\le j,k\le n}$ completely, where $$m_{jk}=\begin{cases}(1+ζ^{j-k})/(1-ζ^{j-k})&\text{if}\ j\not=k,\\δ&\text{if}\ j=k. \end{cases}$$
