The sign of a quasi-Vandermonde determinant is as  

$\det\left(\left[\begin{array}{cccc} \lambda_{1}^{k_{1}} & \lambda_{1}^{k_{2}} & \cdots & \lambda_{1}^{k_{n}}\\ \lambda_{2}^{k_{1}} & \lambda_{2}^{k_{2}} & \cdots & \lambda_{2}^{k_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ \lambda_{n}^{k_{1}} & \lambda_{n}^{k_{2}} & \cdots & \lambda_{n}^{k_{n}} \end{array}\right]\right)>0" original="http://latex.codecogs.com/gif.latex?\det\left(\left[\begin{array}{cccc} \lambda_{1}^{k_{1}} & \lambda_{1}^{k_{2}} & \cdots & \lambda_{1}^{k_{n}}\\ \lambda_{2}^{k_{1}} & \lambda_{2}^{k_{2}} & \cdots & \lambda_{2}^{k_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ \lambda_{n}^{k_{1}} & \lambda_{n}^{k_{2}} & \cdots & \lambda_{n}^{k_{n}} \end{array}\right]\right)>0" style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;$  

where $k_{1}The above result was given and proven, in Apr. 2017, and then is used in the volume computing of a class of special polyhedrons