TITLE: Fractional moments of Poisson distribution
QUESTION [5 upvotes]: I wonder if there is a formula to calculate the fractional (0,1) moments of a Poisson distribution?
Thanks in advance.

REPLY [2 votes]: Fractional factorial moments $M_q$ and cumulants $C_q$ of the Poisson distribution are calculated in Fractional moments of distributions (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 $\lambda=1$.)


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