A novel permanent identity with applications

Abstract

Let $n$ be a positive integer, and define the rational function $S(x_1,\ldots,x_{2n})$ as the permanent of the matrix $[x_{j,k}]_{1\le j,k\le 2n}$, where $$x_{j,k}=\begin{cases}(x_j+x_k)/(x_j-x_k)&\text{if}\ j\not=k,\\1&\text{if}\ j=k.\end{cases}$$ We give an explicit formula for $S(x_1,\ldots,x_{2n})$ which has the following consequence: If one of the variables $x_1,\ldots,x_{2n}$ takes zero, then $S(x_1,\ldots,x_{2n})$ vanishes, i.e., $$\sum_{\tau\in S_{2n}}\prod_{j=1\atop \tau(j)\not=j}^{2n}\frac{x_j+x_{\tau(j)}}{x_j-x_{\tau(j)}}=0,$$ where we view an empty product $\prod_{i\in\emptyset}a_i$ as $1$. As an application, we show that if $\zeta$ is a primitive $2n$-th root of unity then $$\sum_{\tau\in S_{2n}}\prod_{j=1\atop \tau(j)\not=j}^{2n}\frac{1+\zeta^{j-\tau(j)}}{1-\zeta^{j-\tau(j)}}=((2n-1)!!)^2$$ as conjectured by Z.-W. Sun.