# Any way to find a closed form expression, or to simplify this summation?

While solving a certain physical problem, I have reached the following summation:

$$S = \sum_{k = 0}^n \left({n-k-\frac{1}{2} \choose n-k}\right)^2 \ \ \frac{(k+l)!}{k!}$$

where $$n, k$$ and $$l$$ are all non-negative integers. The sum over $$k$$ (as can be seen from the above equation) has to be computed for specified $$(n, l)$$ values.

While it is possible to evaluate this summation for specified $$n$$ and $$l$$ values, using software packages, I was wondering if there is any way to use some $$n \choose k$$ and factorial properties to analytically reduce this expression to a simpler form, ideally even a closed form expression in terms of $$n$$ and $$l$$.

Is that possible, or is this already the simplest possible form of this summation, not reducible any further?

$$\left( {n-1/2\choose n} \right) ^{2}l!\, {\mbox{_3F_2}(-n,-n,l+1;\,-n+1/2,-n+1/2;\,1)}$$
• Thanks Robert. I was guessing this would end up in a ${\ }_3F_2$ function from the background to this summation itself, but somehow, with my scanty mathematical knowledge about these hypergeometric functions, I was reluctant to accept this, and was looking for any possible alternative. Now, further questions - 1) Since the parameters inside the ${\ }_3F_2$ function are repeating, does this ${\ }_3F_2$ function reduce to something simpler for repeated arguments? Any references in this regard? 2) Is it possible to trace the steps from my summation to your closed form expression? Thanks. – 01000100 01010000 May 10 '19 at 16:42