The distributions obtained by left-truncating at k a mixed Poisson distribution, kT-MP, and those obtained by mixing previously left-truncated Poisson distributions, M-kTP, are characterized by means of their probability generating function. The main consequence is that every kT-MP distribution is a M-kTP distribution, but not the other way around.
Valero, J.; Perez, M.; Ginebra, J. On left-truncating and mixing poisson distributions. "Communications in statistics. Theory and methods", 01 Febrer 2014, p. 1-25.