Let $z_{1},\dots,z_{k}$ be distinct complex numbers with $\left|z_{j}\right|=1,\;j=1,\dots,k$. For any natural $N\geqslant k$ consider the rectangular Vandermonde matrix $$ V_{N}=\begin{pmatrix}1 & 1 & \dots & 1\\ z_{1} & z_{2} & & z_{k}\\ \vdots & \vdots & & \vdots\\ z_{1}^{N-1} & z_{2}^{N-1} & & z_{k}^{N-1} \end{pmatrix}. $$

Let $V_{N}^{*}$ denote the conjugate transpose of $V_{N}$. Since $V_{N}$ has full rank, the square $k\times k$ matrix $V_{N}^{*}V_N$ is nonsingular. We are interested in the quantity $$ D\left(N\right)=\det\left(V_{N}^{*}V_N\right). $$ For $N=k$ we have by the well-known explicit formula $$ D\left(k\right)=\prod_{i<j}\left|z_{i}-z_{j}\right|^{2}. $$

**Question: Does there exist an explicit ``simple'' expression
for $D\left(N\right)$ with arbitrary $N>k$?**

Example of a simple expression in the special case $k=2$:

Without loss of generality $z_{1}=1$ and $z_{2}=\exp\left(\imath x\right)$ for $x\in\left[-\pi,\pi\right]\setminus\{0\}.$ An explicit computation gives the following: $$ D\left(N\right)=N^{2}-\frac{\sin^{2}\frac{N}{2}x}{\sin^{2}\frac{x}{2}}. $$