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It seems numerically that this equality holds for $n_1,n_2 \in \mathbb{N}$ :

$\frac{\Gamma (n_1+1) \Gamma (n_2+1)}{\Gamma (n_1+n_2+2)} = \frac{1}{2^{n_1+n_2+2}} \left(\Gamma (n_1+1) \, _2F_1^{(reg)}\left(1,n_1+n_2+2;n_1+2;\frac{1}{2}\right)+\Gamma (n_2+1) \, _2F_1^{(reg)}\left(1,n_1+n_2+2;n_2+2;\frac{1}{2}\right)\right)$

But I don't manage to show it using the properties of the regularized hypergeometric function.

Any help would be welcomed, thanks !

Louka
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  • Usually I can't recognize equations on the spot but I think I actually derived something very similar to this a couple weeks back. Lemme check my notes. If I recall, it had something to do with partial sums of rows of pascal's triangle – WhiteStoneJazz Jan 20 '23 at 09:39
  • Yeah see the answer here: https://math.stackexchange.com/questions/69532/partial-sum-of-rows-of-pascals-triangle So, rows of pascals triangle are related to the HGF which means that any formula you have relating two rows of pascal's triangle (and there are loads) is also a formula relating two HGF's, like the one you found. I'd bet it's a pascal's triangle formula under the hood – WhiteStoneJazz Jan 20 '23 at 09:44
  • I'll try to dive into this path, thanks a lot : – Louka Jan 20 '23 at 13:31

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