Poisson variable X possess characteristic function
$$ \phi_X(t) = E[e^{itX}] = e^{\lambda(z-1)} $$
where $ z = e^{it} $. If i were to obtain the Padé approximation by defining $1/e^{u} = e^{\lambda(z-1)} $ and obtaining $$ e^{u} \approx \frac{1+1/2 u}{1-1/2 u} $$ $$ e^{\lambda(z-1)} \approx -\frac{(2/\lambda-1)+z}{-(2/\lambda+1)+z} $$
I should be able to inverse transform to an approximate pdf in time domain pdf, no?
But requiring poles < 1 for stable decay leads to $ 2/ \lambda +1 < 1$ which obviously is never the case. Where did I think wrong?
In this way I was thinking to obtain distribution for serial correlated Poisson process.