一个类范德蒙矩阵的行列式值估计(整理中)

    若 $0\leq k_{2}

$\det\left(\left[\begin{array}{cccc} 1 & \lambda_{1}^{k_{2}} & \cdots & \lambda_{1}^{k_{n}}\\ 1 & \lambda_{2}^{k_{2}} & \cdots & \lambda_{2}^{k_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ 1 & \lambda_{n}^{k_{2}} & \cdots & \lambda_{n}^{k_{n}} \end{array}\right]\right)=h_{\lambda_{1},\lambda_{2},\cdots,\lambda_{n}}^{(s)}\prod_{1\leq j_{1}0$

其中 $s=\sum_{j=2}^{n}\left(k_{j}-j+1\right)$ , $h_{\lambda_{1},\lambda_{2},\cdots,\lambda_{n}}^{(s)}$ 为非零的如下关于 $n$ 元变量 $\lambda_{i}(i=\overline{1,n})$ 的 $s$ 次的齐次多项式

$h_{\lambda_{1},\lambda_{2},\cdots,\lambda_{n}}^{(s)}=\sum_{\sum_{j=1}^{n}s_{j}=s}g_{s_{1},s_{2},\cdots,s_{n}}\prod_{j=1}^{n}\lambda_{j}^{s_{j}}$

其中所有 $g_{s_{1},s_{2},\cdots,s_{n}}\geq0$ 且至少存在某个 $g_{s_{1},s_{2},\cdots,s_{n}}>0$ .