Fractional <A HREF="https://en.wikipedia.org/wiki/Factorial_moment">factorial moments</A> $M_q$ and cumulants $C_q$ of the Poisson distribution are calculated in <A HREF="http://www.jetpletters.ac.ru/ps/1310/article_19793.pdf">Fractional moments of distributions</A> (1994)$^\ast$    

$$M_q=e^{-\lambda}\frac{\Phi(1,1-q;\lambda)}{\Gamma(1-q)},$$
$$C_q=\frac{q\lambda}{\Gamma(2-q)},$$

where $\Phi$ is the confluent hypergeometric function. For integer $q=n$ one recovers the usual results 
$$M_n=\lambda^n,\;\;C_n=\lambda\delta_{n,1},$$ 
in view of the limit
$$\lim_{q\rightarrow n}\frac{\Phi(1,1-q;\lambda)}{\Gamma(1-q)}=e^\lambda \lambda^n.$$

In the interval $n<q<n+1$ both $M_q$ and $C_q$ oscillate and can become negative. (The plot shows $M_q$ for $\lambda=1$.)

<IMG SRC="https://ilorentz.org/beenakker/MO/fractionalmoment.png"/>

---
$^\ast$<sub>There is a typo in equation (7) of the cited paper, the denominator $\langle n\rangle$ should be $\langle n\rangle^q$.</sub>