Displaced Poisson Distribution

In statistics, the displaced Poisson, also known as the hyper-Poisson distribution, is a generalization of the Poisson distribution.
The probability mass function is

:
P(X=n) = \begin
e^\dfrac\cdot\dfrac, \quad n=0,1,2,\ldots &\text r\geq 0\\ 0pt e^\dfrac\cdot\dfrac,\quad n=s,s+1,s+2,\ldots &\text
\end

where $\lambda>0$ and ''r'' is a new parameter; the Poisson distribution is recovered at ''r'' = 0. Here $I\left(r,\lambda\right)$ is the Pearson's incomplete gamma function:
:$I(r,\lambda)=\sum^\infty_\frac,$
where ''s'' is the integral part of ''r''.
The motivation given by Staff is that the ratio of successive probabilities in the Poisson distribution (that is $P(X=n)/P(X=n-1)$) is given by $\lambda/n$ for $n>0$ and the displaced Poisson generalizes this ratio to $\lambda/\left(n+r\right)$.

References

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Discrete distributions