I came across a sum of binomial coefficients while trying to solve a problem involving $SU(2)$ group integrals. I am not able to solve it, nor I found a similar identity in the literature. I would like to ask if anyone knows of any techniques/references that I might use in order to find a solution. The expression is the following

$$ \sum_{i=\max(0,p)}^{\min(n+p,m)} \sum_{i'=\max(0,p')}^{\min(n'+p',m')} \binom{n+p}{i} \binom{n'+p'}{i'} \binom{i-p+m'-i'}{m'-i'} \binom{i'-p'+m-i}{m-i} =F(m,n,p|m',n',p') $$

It is very symmetric, and that gives me hope to find a solution, even though it depends on 6 independent non-negative integers, $m,n,p$ and $m',n',p'$. (**Edit:** in fact it should be $p\geq-n$ and $p'\geq-n'$).

I noticed that in most (if not all) binomial identities, there are more instances of the summation index (e.g. $i$) on the lower part of the binomial coefficients than on the top, in any given identity. Here the situation is the opposite, and it is the reason why I cannot solve this expression.

Thanks!

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