I'm trying to solve a problem. It led to such a summation :
$$f(n)=\sum_{i\, =\, \left\lceil\, n/2 \,\right\rceil}^{\infty} {\binom{i + k}{i}\binom{2i}{n}{(-a)}^i}$$
Where $k$ is a nonnegative integer, $a$ is a non-negative real number and both are given.
How can I find it as a closed form function of $n$?
I am not sure that it coud simplify since it is almost the definition of the gaussian hypergeometric function.
If $n=2m$, you would get $$(-a)^m \binom{k+m}{m} \, _2F_1\left(m+\frac{1}{2},k+m+1;\frac{1}{2};-a\right)$$