On determinants involving $(\frac{j+k}{p})\pm (\frac{j-k}{p})$

Abstract

Let $p=2n+1$ be an odd prime. In this paper, we mainly evaluate determinants involving $(\frac{j+k}{p})\pm (\frac{j-k}{p})$, where $(\frac{\cdot }{p})$ denotes the Legendre symbol. When $p\equiv 1(\mathrm{mod}4)$, we determine the characteristic polynomials of the matrices $${\left[\left(\frac{j+k}{p}\right)+\left(\frac{j-k}{p}\right)\right]}_{1\le j,k\le n}\text{and}{\left[\left(\frac{j+k}{p}\right)-\left(\frac{j-k}{p}\right)\right]}_{1\le j,k\le n},$$ and also establish the general identity \begin{align*} &\ \left|x+\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)+\left(\frac jp\right)y+\left(\frac kp\right)z+\left(\frac{jk}p\right)w\right|_{1\le j,k\le n} \\=&\ (-p)^{(p-5)/4}\left(\left(\frac{p-1}2\right)^2wx-\left(\frac{p-1}2y-1\right)\left(\frac{p-1}2z-1\right)\right). \end{align*}

16 pages. Make Conjecture 1.1 stronger