On some determinants involving the tangent function

Abstract

Let $p$ be an odd prime and let $a,b\in \mathbb{Z}$ with $p\nmid ab$. In this paper we mainly evaluate $$T_p^{(\delta )}(a,b,x):=\det {\left[x+\tan \pi \frac{a{j}^2+b{k}^2}{p}\right]}_{\delta \le j,k\le (p-1)/2}(\delta =0,1).$$ For example, in the case $p\equiv 3(\mathrm{mod}4)$ we show that $T_p^{(1)}(a,b,0)=0$ and $$T_p^{(0)}(a,b,x)=\{\begin{array}{ll}{2}^{(p-1)/2}{p}^{(p+1)/4}& \text{if}(\frac{ab}{p})=1,\\ p^{(p+1)/4}& \text{if}(\frac{ab}{p})=-1,\end{array}$$ where $(\frac{\cdot }{p})$ is the Legendre symbol. When $(\frac{-ab}{p})=-1$, we also evaluate the determinant $\det [x+\cot \pi \frac{a{j}^2+b{k}^2}{p}{]}_{1\le j,k\le (p-1)/2}.$ In addition, we pose several conjectures one of which states that for any prime $p\equiv 3(\mathrm{mod}4)$ there is an integer $x_p\equiv 1(\mathrm{mod}p)$ such that $$\det {\left[\sec 2\pi \frac{(j-k{)}^2}{p}\right]}_{0\le j,k\le p-1}=-p^{(p+3)/2}{x}_p^2.$$

20 pages. To appear in Ramanujan J

Cited by in corpus (1)

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