While trying to calculate a discrete probability distribution involving the composition of two Poisson and two Binomial distributions, I keep ending up in the following term:

$$ \sum_{m = 0}^M A^m \binom{M}{m} \binom{N-M}{n-m} $$

where $M$, $n$ and $N$ are integer ($M$ is summed up later on, the other two are the variables of my pmf) and $A$ is a positive real number coming from a combination of the elementary probability of the two binomials.

With respect to the Chu-Vandermonde identity, the extra power term makes the mess and I cannot simplify the expression, assuming that a sum expression exists at all.

It seems that there is a sum expression when $A=-1$, and I tried to use the same approach as Sum of product of two binomial coefficients: indeed this "coefficient of"-operator helps to visualize the steps, however in the last step I just arrive back to the initial expression.

Any hint, or even a confirmation that a sum expression does not exist, is very welcome.

P.S. I can write down the full problem if this background information is needed


According to Maple, it "simplifies" to a hypergeometric: $${N-M\choose n}{\mbox{$_2$F$_1$}(-n,-M;\,N-M-n+1;\,A)}$$ For this to exist I think you want $N-M-n+1 \ge 0$.

  • $\begingroup$ I suspected why the quotes for "simplifies", and after reading a bit about this hypergeometric functions it is clear. Well, I like it or not, I guess this answer my question $\endgroup$
    – Matteo
    Mar 29 '20 at 22:20

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