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I want to calculate a sum of the form $$\sum_{k=0}^m \frac{\Gamma[m+1+\alpha-k]^2}{\Gamma[m+1-k]^2}\frac{\Gamma[x+k]}{\Gamma[x]k!}$$ where $m>0$ and belongs to integers and $\alpha$ takes half integer values. Is there any closed form expression for the sum? I tried to do the sum but it gave me $$\, _3F_2\left(-m,-m,x;-\alpha-m,-\alpha-m;1\right)$$ with some other overall factors. So either I need a closed form expression for the sum or the closed form expression for the hypergeometric function.

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