类似拉格朗日插值的恒等式

青青子衿

这一步是显然的吗？下述恒等式有没有推广？😮
\begin{align*}
I&=\sum _{k=1}^n \frac{1}{\left[\frac{\partial }{\partial t}\prod _{i=1}^n\left(t-x_i\right)\right]_{t=x_k}}\left[\dfrac{\partial }{\partial t}\frac{\prod _{i=1}^{2n+1} \left(t-y_i\right)}{\left(t-y_j\right)^2\frac{\partial }{\partial t}\prod _{i=1}^{n} \left(t-x_i\right)}\right]_{t=x_k}\\
&=1-\dfrac{1}{\prod _{i=1}^{n} \left(y_j-x_i\right)^2}\left[\frac{\partial }{\partial t}\prod _{i=1}^{n} \left(t-y_i\right)\right]_{t=y_j}
\end{align*}