On generalized Legendre matrices involving roots of unity over finite fields

In this paper, motivated by the work of Chapman, Vsemirnov and Sun et al., we investigate some arithmetic properties of the generalized Legendre matrices over finite fields. For example, letting $a_1,\cdots ,a_{(q-1)\mathrm{/}2}$ be all non-zero squares in the finite field ${\mathbb{F}}_q$ which contains $q$ elements with $2\nmid q$, we give the explicit value of $D_{(q-1)\mathrm{/}2}=\det [(a_i+a_j{)}^{(q-3)\mathrm{/}2}{]}_{1\le i,j\le (q-1)\mathrm{/}2}$. In particular, if $q=p$ is a prime greater than $3$, then $$\left(\frac{\det D_{(p-1)\mathrm{/}2}}{p}\right)=\{\begin{array}{lll}1& ifp\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4),(-1{)}^{(h(-p)+1)\mathrm{/}2}& ifp\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)\text{ and }p>3,\end{array}$$ where $(\cdot \mathrm{/}p)$ is the Legendre symbol and $h(-p)$ is the class number of $\mathbb{Q}(\sqrt{-p})$.