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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?

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Maple writes this using a hypergeometric function:

$$ \left( {n-1/2\choose n} \right) ^{2}l!\, {\mbox{$_3$F$_2$}(-n,-n,l+1;\,-n+1/2,-n+1/2;\,1)} $$

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  • $\begingroup$ 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. $\endgroup$ May 10, 2019 at 16:42

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