dhermite,
distribution function phermite, quantile function
qhermite and random generation rhermite for the
generalized Hermite distribution. The probability mass function is usually
parametrized in terms of the mean $\mu$ and the index of dispersion
$d = \frac{\sigma^2}{\mu}$:$P(X=x) = P(X=0) \frac{\mu^x (m-d)^x}{(m-1)^x} \sum_{j=0}^{[x/m]}
\frac{(d-1)^j (m-1)^{(m-1)j}}{m^j \mu^{(m-1)j} (m-d)^{mj} (x-mj)!j!}$
where $P(X=0) = exp(\mu (-1+ \frac{d-1}{m}))$, m is the degree of
the generalized Poisson distribution and $[x/m]$ is the integer part of
$x/m$.
McKendrick A G Applications of Mathematics to Medical Problems. Proceedings of the Edinburgh Mathematical Society 1926;44:98–130.
Kemp A W, Kemp C D. An alternative derivation of the Hermite distribution. Biometrika 1966;53 (3-4):627–628.
Patel Y C. Even Point Estimation and Moment Estimation in Hermite Distribution. Biometrics 1976;32 (4):865–873.
Gupta R P, Jain G C. A Generalized Hermite distribution and Its Properties. SIAM Journal on Applied Mathematics 1974;27:359–363.
Bekelis, D. Convolutions of the Poisson laws in number theory. In Analytic & Probabilistic Methods in Number Theory: Proceedings of the 2nd International Conference in Honour of J. Kubilius, Lithuania 1996;4:283–296.
Zhang J, Huang H. On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies. American Journal of Theoretical and Applied Statistics 2013;2:110–121.
Kotz S. Encyclopedia of statistical sciences. John Wiley 1982-1989.
Kotz S. Univariate discrete distributions. Norman L. Johnson 2005.
Puig P. (2003). Characterizing Additively Closed Discrete Models by a Property of Their Maximum Likelihood Estimators, with an Application to Generalized Hermite Distributions. Journal of the American Statistical Association 2003; 98:687–692.
Distributions for some other distributions,
qhermite, phermite, rhermite,
hermite-package, glm.hermite