Fractional <A HREF="https://en.wikipedia.org/wiki/Factorial_moment">factorial moments</A> $F_q$ and cumulants $K_q$ 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):

$$F_q=e^{-\lambda}\lambda^{q-1}\frac{\Phi(1,1-q,\lambda)}{\Gamma(1-q)},$$
$$K_q=q\lambda\Gamma(2-q),$$

where $\Phi$ is the confluent hypergeometric function. For integer $q$ one recovers the usual results $F_q=\lambda^q$ and $K_q=\lambda\delta_{q,1}$. In the interval between integer $q$ both $F_q$ and $K_q$ oscillate.