On determinants involving $(\frac{j^2-k^2}p)$ and $(\frac{jk}p)$

Abstract: Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. In this paper, we study the determinant $$\det\left[\left(\frac{j^2-k^2}p\right)+\left(\frac{jk}p\right)w\right]_{\delta\le j,k\le (p-1)/2}$$ with $\delta\in{0,1}$. For example, we prove that the determinant does not depend on $w$ if $p\equiv3\pmod4$ and $\delta=0$.