On some determinants involving the tangent function

Let p be an odd prime and let \(a,b\in {\mathbb {Z}}\) $a,b\in \mathbb{Z}$ with \(p\not \mid ab\) $p\nmid ab$ . In this paper,we mainly evaluate \(\begin{aligned} T_p^{(\delta )}(a,b,x):=\det \left[ x+\tan \pi \frac{aj^2+bk^2}{p}\right] _{\delta \leqslant j,k\leqslant (p-1)/2}\ \ (\delta =0,1). \end{aligned}\) $$\begin{array}{c}\hfill T_p^{(\delta )}(a,b,x):=det{\left[x,+,tan,\pi ,\frac{a{j}^2+b{k}^2}{p}\right]}_{\delta ⩽j,k⩽(p-1)/2}(\delta =0,1).\end{array}$$ For example, in the case \(p\equiv 3\ ({\textrm{mod}}\ 4)\) $p\equiv 3(\text{mod}4)$ , we show that \(T_p^{(1)}(a,b,0)=0\) $T_p^{(1)}(a,b,0)=0$ and \(\begin{aligned} T_p^{(0)}(a,b,x)={\left\{ \begin{array}{ll} 2^{(p-1)/2}p^{(p+1)/4}&{}\text {if}\ (\frac{ab}{p})=1, \\ p^{(p+1)/4}&{}\text {if}\ (\frac{ab}{p})=-1,\end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill T_p^{(0)}(a,b,x)=\left\{\begin{array}{cc}{2}^{(p-1)/2}{p}^{(p+1)/4}\hfill & \text{if}(\frac{\mathit{ab}}{p})=1,\hfill \\ p^{(p+1)/4}\hfill & \text{if}(\frac{\mathit{ab}}{p})=-1,\hfill \end{array}\right)\end{array}$$ where \((\frac{\cdot }{p})\) $(\frac{·}{p})$ is the Legendre symbol. When \((\frac{-ab}{p})=-1\) $(\frac{-ab}{p})=-1$ , we also evaluate the determinant \(\det [x+\cot \pi \frac{aj^2+bk^2}{p}]_{1\leqslant j,k\leqslant (p-1)/2}.\) $det{[x+cot\pi \frac{a{j}^2+b{k}^2}{p}]}_{1⩽j,k⩽(p-1)/2}.$ In addition, we pose several conjectures one of which states that for any prime \(p\equiv 3\ ({\textrm{mod}}\ 4)\) $p\equiv 3(\text{mod}4)$ , there is an integer \(x_p\equiv 1\ ({\textrm{mod}}\ p)\) $x_p\equiv 1(\text{mod}p)$ such that \(\begin{aligned}\det \left[ \sec 2\pi \frac{(j-k)^2}{p}\right] _{0\leqslant j,k\leqslant p-1}=-p^{(p+3)/2}x_p^2.\end{aligned}\) $$\begin{array}{c}\hfill det{\left[sec,2,\pi ,\frac{{(j-k)}^2}{p}\right]}_{0⩽j,k⩽p-1}=-p^{(p+3)/2}{x}_p^2.\end{array}$$