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Probability question

I've attached a PDF outlining some probability calculations for the boardgame Risk I am working on.

My current problems occur with equation (5). I would like to figure out how to write it as a closed-form function (or 2 closed-form functions, one for L is less than 0 and one for the opposite) of m, n, M, N, s, L (L=(m-n)-(M-N) as given), the f transition probabilities, and the dummy variable k. My difficulties with doing this are: 1) what is this δm,M+N-n-2s in closed form? and 2) how can we write the summation bound in a closed-form manner that can be used to calculate.
Risk Probability Distributions, Daniel Taflin.pdf247.5KB
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