# sum over combination of binomial coefficients similar to Vandermonde identity, but with extra power series factor

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: $${N-M\choose n}{\mbox{_2F_1}(-n,-M;\,N-M-n+1;\,A)}$$ For this to exist I think you want $$N-M-n+1 \ge 0$$.