I have a larger expression that should simplify to some closed-form solution, but I can't figure out how (and neither does mathematica). A crucial part in that larger expression is the following sum,
$$ \sum_{j=0}^{\mathrm{min}(a,b)} \binom{a}{j} \, \binom{a+b-j-1}{a-1} x^j, $$
where $a \ge 2$, $b \ge 1$ integers and $x$ real. Does anyone know of a different / simpler expression for this? Seems related to generating functions involving binomial coefficients (wiki). Any help really appreciated!
This is a hypergeometric function (specifically, it is $\binom{a + b - 1}{a - 1} \cdot {_2F_1}(a, b; a + b - 1; x)$) and in general it does not factor as a product nicely (you can probably convince yourself of this by plugging in any particular values). However, that doesn't mean some larger expression containing it couldn't factor or otherwise be nice.