There's a debate with some colleagues about the meaning of the notation $\sum_{(i,m) \neq (j,n)}$. Which one of the following is correct?
1) $\sum_{(i,m) \neq (j,n)} a_{i} b_{m} = \sum_{i} \sum_{m} a_{i} b_{m} - a_{j} b_{n}$
2) $\sum_{(i,m) \neq (j,n)} a_{i} b_{m} = \sum_{i \neq j} \sum_{m \neq n} a_{i} b_{m}$
I tend to say that the first one is correct since $(i,m) \neq (j,n)$ should be referred to the pair and not the single indices.
The meaning of
$$\sum_{(i,m)\neq (j,n)} a_ib_m$$
is "the sum over all values $i,m$, except for the pair $(j,n)$." So the first sum is the correct one. Alternatively, you could write the same expression as
$$\sum_{i\neq j}\sum_{m}a_i b_m + \sum_{m\neq n} a_jb_m$$
Your second expression also excludes all values of the form $(i,n)$ from the sum, which is not correct.