A novel permanent identity with applications

Abstract

Let $n$ be a positive integer, and define the rational function $S(x_1,\dots ,x_{2n})$ as the permanent of the matrix $[x_{j,k}{]}_{1\le j,k\le 2n}$, where $$x_{j,k}=\{\begin{array}{ll}(x_j+x_k)/(x_j-x_k)& \text{if}j\ne k,\\ 1& \text{if}j=k.\end{array}$$ We give an explicit formula for $S(x_1,\dots ,x_{2n})$ which has the following consequence: If one of the variables $x_1,\dots ,x_{2n}$ takes zero, then $S(x_1,\dots ,x_{2n})$ vanishes, i.e., $$\sum _{\tau \in S_{2n}}\prod _{\genfrac{}{}{0ex}{}{j=1}{\tau (j)\ne j}}^{2n}\frac{x_j+x_{\tau (j)}}{x_j-x_{\tau (j)}}=0,$$ where we view an empty product ${\prod }_{i\in \varnothing }{a}_i$ as $1$. As an application, we show that if $\zeta$ is a primitive $2n$-th root of unity then $$\sum _{\tau \in S_{2n}}\prod _{\genfrac{}{}{0ex}{}{j=1}{\tau (j)\ne j}}^{2n}\frac{1+{\zeta }^{j-\tau (j)}}{1-{\zeta }^{j-\tau (j)}}=((2n-1)!!{)}^2$$ as conjectured by Z.-W. Sun.

15 pages, expanded version with new results added