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I have the following sum \begin{equation} \sum_{r_1=q+1}^{\tau}\dots\sum_{r_\lambda=q+1}^{\tau}{\tau\choose r_1,\dots,r_\lambda,\tau-r_1-\dots -r_\lambda} (\Lambda-\lambda)^{\tau-r_1-\dots-r_\lambda} \end{equation} and would like to estimate its value. Any ideas or techniques to do this? By the way, it is not always the case that $\tau>>q$, so the entire range of possibilities is necessary.

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is $\lambda$ fixed positive integer and $\Lambda>\lambda$ fixed real? – Fedor Petrov Nov 9 '10 at 11:45
yes to both. In addition, $\Lambda$, $\lambda$, $q$ and $\tau$ are also integers. I realize that using incomplete beta functions, for the case of $\lambda=1$, one can obtain a closed form expression, and for $\lambda=2$ then the incomplete beta function helps re-write the first sum, but then ends up inside the second sum. However, this is as far as I have gotten. – Eduardo Lopez Nov 9 '10 at 12:42
I've found one reference that is quite useful: Levin, Ann Stat 9, 1123 (1981) for the cumulative of multinomial distributions. – Eduardo Lopez Nov 9 '10 at 15:39
Also, try a search of the questions and answers here in MO: there are a number of them about sum of multinomial coefficients close to yours. – Pietro Majer Nov 9 '10 at 15:46
As written the multinomial can have negative arguments. Is this intended? – Steve Huntsman Nov 9 '10 at 15:55

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