On recovering a mixed Poisson distribution from its left-truncated version

An algorithm is proposed that, starting from the probability generating function of a left-truncation at k of a mixed Poisson distribution, recovers the first k+1 probabilities of the untruncated distribution, without the need of eliciting what the mixing distribution is. The result establishes that irrespective of the value where the distribution is truncated, there still remains enough information in the tail so that the initial mixing distribution can be recovered.