A minimum variance unbiased estimator is provided for the parameter of a truncated Poisson distribution in the case of truncation on the left. In this connection the distribution is obtained for the sum of $n$ independent identically distributed truncated Poisson random variables, and then well-known properties of sufficient statistics are employed to obtain the estimator. For the case of truncation away from the zero value results are expressed in terms of Stirling numbers of the second kind. The estimator has a particularly simple form and tables are available for its computation. For the general case results are expressed in terms of what we call generalized Stirling numbers. As a by-product of the statistical considerations there arises an identity between generalized Stirling numbers which may be useful in the study of Difference Equations.
"Minimum Variance Unbiased Estimation for the Truncated Poisson Distribution." Ann. Math. Statist. 29 (3) 755 - 765, September, 1958. https://doi.org/10.1214/aoms/1177706534