What is please the Laplace transform (moment generating function $M(t)$) of a generalised hypergeometric distribution shown below $$p_X(x)=K\cdot\frac{(a_1)_x\dots(a_p)_x}{(b_1)_x\dots(b_q)_x}\cdot\frac{\theta^x}{x!}$$ for $\theta>0$ and some constant $K$?
Meaning $$\mathcal{L}\left\{ p_X(x)\right\} \left(s\right)=K\cdot\sum_{x\geq0}\frac{(a_1)_x\dots(a_p)_x}{(b_1)_x\dots(b_q)_x}\cdot\frac{\theta^x}{x!}e^{-xs} $$ In the case you would know also the cumulant generating function $K(t)=\log M(t)$, I would be most obliged.