Fractional factorial moments $F_q$ and cumulants $K_q$ of the Poisson distribution are calculated in Fractional moments of distributions (1994):
$$F_q=e^{-\lambda}\frac{\Phi(1,1-q,\lambda)}{\Gamma(1-q)},$$ $$K_q=q\lambda\Gamma(2-q),$$
where $\Phi$ is the confluent hypergeometric function. For integer $q=n$ one recovers the usual results $F_n=\lambda^n$ and $K_n=\lambda\delta_{n,1}$, in view of the limit $$\lim_{q\rightarrow n}\frac{\Phi(1,1-q,\lambda)}{\Gamma(1-q)}=\lambda^n.$$
In the interval between integer $q$ both $F_q$ and $K_q$ oscillate.