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

Let $p$ be an odd prime and let $(\frac{\cdot }{p})$ be the Legendre symbol. In this paper, we study the determinant $$\det {[\left(\frac{j^2-k^2}{p}\right)+\left(\frac{jk}{p}\right)w]}_{\delta \le j,k\le (p-1)\mathrm{/}2}$$ with $\delta \in \{0,1\}$. For example, we prove that the determinant does not depend on $w$ if $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and $\delta =0$.