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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.

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

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 $\langle n\rangle^q$.}

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.

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.

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

Fractional factorial moments $F_q$ and cumulants $K_q$ of the Poisson distribution are calculated in Fractional moments of distributions (1994):

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

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.

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.