I need to evaluate the following finite sum:


$$
\sum\_{j=0}^{h}(-1)^j\binom{h}{j}(jx)\_{k},\qquad k\geq h,\, x\in\mathbb{R}^{+}
$$


and


$$
(jx)\_{k}=jx(jx-1)\cdots(jx-k+1)
$$
is the falling factorial. Any hints would be appreciated!



Finite sum with falling factorial
This sum is $k!$ times the coefficient of $z^k$ in $[1-(1+z)^x]^h$. So it can be simplified for a few values of $x$ (specifically $x=0$, $1$, $-1$, $2$, or $1/2$) but not in general.


