2
$\begingroup$

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

$\endgroup$
2
$\begingroup$

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 $N-M-n+1 \ge 0$.

$\endgroup$
1
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.