A Poisson distribution with mean $\lambda$ for a random variable X is given by the probability distribution
$P(X=k) = \mathcal{P}_\lambda(k) = \lambda^{k}\ \frac{e^{-\lambda}}{k!} \,.$
Let this be the conditional distribution $P(X=k|\lambda)$. Then we can encounter scenarios (constant priors, not sure if this should be reconsidered in this context) where through Bayes theorem, one gets
$P(\lambda|X=k) \propto P(X=k|\lambda) = \mathcal{P}_\lambda(k)\,.$
However, this is not a Poisson distribution anymore, since $\lambda$ is the random variable now. Are there any analytically known results about this distribution that arises from swapping variables in the Poisson distribution?
Besides the usual mean and variance, I am particularly looking for the corresponding characteristic function and, if possible, composition laws for sums and differences of such variables (such as they exist for the Poisson distribution itself, see Poisson Distribution of sum of two random independent variables $X$, $Y$, Sum of two independent Skellam distributed random variables).
