Looking for some help with a proof by iduction. Im looking to proove the following summation holds true:
$$\frac{\langle W \vert (C)^{N} \vert V \rangle}{\langle W \vert \vert V \rangle}=\frac{\langle W \vert (D+E)^{N} \vert V \rangle}{\langle W \vert \vert V \rangle}= \sum_{p=0}^N\frac{p(2N-1-p)!}{N!(N-p)!}\frac{\beta^{-p-1}-\alpha^{-p-1}}{\beta^{-1}-\alpha^{-1}}$$
The problem is related to the following rules: $$DE= D+E=C $$
$$D\vert V \rangle= \frac{1}{\beta} \vert V \rangle $$
$$\langle W\vert E= \frac{1}{\alpha}\langle W \vert $$
Now using the base case of N=1 yields to, $\frac{1}{\alpha}+\frac{1}{\beta}$ which I have shown but struggling with the hypothesis step.
Any help would be most appreciated, many thanks.