I'm pretty sure that this has been asked before but I can't find it anywhere. All my search results are clouded by GCSE revision on expanding brackets.
The Problem:
Fix $m\in\Bbb N$ and $i\in\overline{1, m}$. Expand and simplify $$\left(\sum_{r=1,\\ r\neq i}^mT_r\right)\left(\sum_{s=1,\\ s\neq i}^mT_s^{-1}\right)$$ for non-zero complex numbers $(T_j)_{j\in\overline{1, m}}$.
My Attempt:
We have
$$\begin{align} \left(\sum_{r=1,\\ r\neq i}^mT_r\right)\left(\sum_{s=1,\\ s\neq i}^mT_s^{-1}\right)&=1+T_1T_2^{-1}+\dots +T_1T_{i-1}^{-1}+0+T_1T_{i+1}^{-1}+\dots +T_1T_m^{-1} \\ &+T_2T_1^{-1}+1+\dots +T_2T_{i-1}^{-1}+0+T_2T_{i+1}^{-1}+\dots +T_2T_m^{-1} \\ &+ \\ &\vdots \\ &+T_{i-1}T_1^{-1}+\dots +T_{i-1}T_{i-2}^{-1}+1+0+T_{i-1}T_{i+1}^{-1}+\dots +T_{i-1}T_m^{-1} \\ &+T_{i+1}T_1^{-1}+\dots +T_{i+1}T_{i-1}^{-1}+0+1+T_{i+1}T_{i+2}^{-1}+\dots +T_{i +1}T_m^{-1} \\ &+ \\ &\vdots \\ &+T_mT_1^{-1}+\dots +T_mT_{i-1}^{-1}+0+T_mT_{i+1}^{-1}+\dots +1 \\ &=(m-1)+X, \end{align}$$
but I don't know what that $X$ should be.
I expect to see binomial coefficients in there.
Added Complication:
I would prefer not to relabel. This problem arose in my research, where it's important to keep track on the subscripts.
Please help :)
It's much simpler than I thought:
$$X=\sum_{r=1,\\ r\neq i}^m\sum_{s=1,\\ s\neq i,\\ s\neq r}^m\frac{T_r}{T_s}.$$