$X$ has zero truncated poission distribution ($\lambda$)
then pmf is $f(x)=\frac{\lambda^x}{(e^\lambda-1)x!}$
and the probability generating function is $\phi_X(s)=\frac{e^{\lambda s}-1}{e^\lambda-1}$,
Now i have anthor variable $\epsilon$ has the probability generating function
$\phi_\epsilon(s)=\frac{\phi_X(s)}{e^{-\lambda}+(1-e^{-\lambda})\phi_X(1-\alpha+\alpha s)}=\frac{e^{\lambda s}-1}{(1-e^\lambda)e^{\lambda(1-\alpha+\alpha s)}}$
and then $\epsilon$ has the pmf as $P(\epsilon=j)=\frac{[(1-\alpha)^j-(-\alpha)^j]\lambda^j e^{\lambda \alpha}}{j! (e^\lambda-1)}$...... λ>0 and 0≤α≤1/2
What type of distribution for epsilon? can I generate observations for epsilon ?