Fractional moments of Poisson distribution

Question

I wonder if there is a formula to calculate the fractional (0,1) moments of a Poisson distribution?

Thanks in advance.

Answer 1

Fractional moments of any real order $p$ of any real-valued random variable $X$ with $E|X|^p<\infty$ can be expressed in terms of the characteristic function $f$ of $X$ as follows: \begin{equation} E\overline{X^p}=i^{-p}f^{(p)}(0), \end{equation} where $\overline z$ denotes the complex conjugate of a complex number $z$, $0^0:=0$, $0^p:=\infty$ if $p<0$, and $f^{(p)}$ is the derivative of $f$ of (possibly fractional) order $p$ (note that values of $X^p$ may be complex numbers if $p$ is not an integer and if $X$ may take negative values) -- see e.g. formula (22) in Positive-part moments via the characteristic functions; also, JoTP.

In particular, if $X$ has the Poisson distribution with real parameter $\lambda>0$, then $f(t)=\exp\{\lambda(e^{it}-1)\}$ for real $t$ and hence, by formula (77) in the mentioned paper,
\begin{equation} EX^p=\frac1{i^p\Gamma(-p)}\,\int_0^\infty\frac{\exp\{\lambda(e^{-is}-1)\}-1}{s^{1+p}} \, ds \end{equation} for $p\in(0,1)$. The latter integral apparently cannot be expressed in closed form; at least, Mathematica quickly gives up on it. However, it provides for effective calculations of the fractional moments, as explained in the same paper.

Answer 2

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 $\lambda=1$.)

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