Closed form for $\sum_{j=0}^{\infty}{\alpha \choose j} {\beta \choose j}x^j$

WolframAlpha immediately gives hypergeometric form $F_{\alpha, \beta}(x) = {}_2 F_1(-\alpha, -\beta; 1; x)$.

Here is a "metaproof" that no simple closed form exists.

A conjecture by Carnevale and Voll states that:

For nonnegative integers $\alpha,\beta$ with $\alpha>\beta$, we have that $$ F_{\alpha,\beta}(-1)\neq 0. $$

As far as I know, the conjecture is still open!

For recent work in this direction see this article by Habsieger.