一个特殊矩阵的行列式值

$\mathrm{det}\left(\left[\begin{array}{cccc} \sum_{i=1}^{n}\frac{k_{1}!}{(i-1)!(k_{1}-i+1)!}\lambda^{k_{1}-i+1}b_{i} & \sum_{i=1}^{n}\frac{k_{2}!}{(i-1)!(k_{2}-i+1)!}\lambda^{k_{2}-i+1}b_{i} & \cdots & \sum_{i=1}^{n}\frac{k_{n}!}{(i-1)!(k_{n}-i+1)!}\lambda^{k_{n}-i+1}b_{i}\\ \vdots & \vdots & \cdots & \vdots\\ \lambda^{k_{1}}b_{n-1}+k_{1}\lambda^{k_{1}-1}b_{n} & \lambda^{k_{2}}b_{n-1}+k_{2}\lambda^{k_{2}-1}b_{n} & \ddots & \lambda^{k_{n}}b_{n-1}+k_{n}\lambda^{k_{n}-1}b_{n}\\ \lambda^{k_{1}}b_{n} & \lambda^{k_{2}}b_{n} & \cdots & \lambda^{k_{n}}b_{n} \end{array}\right]\right)$

      $=\mathrm{det}\left(\left[\begin{array}{cccc} \frac{k_{1}!}{(n-1)!(k_{1}-n+1)!}\lambda^{k_{1}-n+1}b_{n} & \frac{k_{2}!}{(n-1)!(k_{2}-n+1)!}\lambda^{k_{2}-n+1}b_{n} & \cdots & \frac{k_{n}!}{(n-1)!(k_{1}-n+1)!}\lambda^{k_{n}-n+1}b_{n}\\ \vdots & \vdots & \cdots & \vdots\\ k_{1}\lambda^{k_{1}-1}b_{n} & k_{2}\lambda^{k_{2}-1}b_{n} & \ddots & k_{n}\lambda^{k_{n}-1}b_{n}\\ \lambda^{k_{1}}b_{n} & \lambda^{k_{2}}b_{n} & \cdots & \lambda^{k_{n}}b_{n} \end{array}\right]\right)$

      $=(-1)^{n-1}\left(\prod_{j=1}^{n}\frac{1}{(n-j)!}\right)\lambda^{\sum_{j=1}^{n}k_{j}-n(n-1)/2}b_{n}^{n}\prod_{1\leq i