How would you write this sum most elegantly? (Or in other words: What is the most elegant and best way to show a sum of products with indicies?)
$\sum_{k} A_{a_{1},...,a_{i},k}*B_{k,b_{1},...,b_{j}}$
This just seems so clunky ...
EDIT:
$R_{m_{1}...m_{i}n_{1}...n_{j}} = \sum_{k} A_{m_{1}...m_{i}k} * B_{kn_{1}...n_{j}}$
This isn't better
Consider using the Einstein summation convention where a repeated index (in your case k) is implicitly understood to be summed over (so one no longer writes the $\Sigma$). Also the * is definitely not needed.
Can you figure out how to write your product in this way?
After that, an even more abstract notation is to to drop the indices on the right hand side altogether, leaving them implied by the left hand side, and represent multiplication with a dot. So the RHS of your edit would look like $A \cdot B$. One can infer from the indices on the LHS and the ranks of the tensors A and B how many indices are contracted (equal and summed over). However, this requires a convention on the ordering of the indices (the last indices of A and the first of B are summed over) and if the tensors A and B have no definite symmetry it may not be practical to impose this.