Fractional factorial moments $F_q$ and fractional factorial cumulants $K_q$ of the Poisson distribution are calculated in Fractional moments of distributions (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. Check that 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.