On discrete distributions of orderk

Summary The class of discrete distributions of orderk is defined as the class of the generalized discrete distributions with generalizer a discrete distribution truncated at zero and from the right away fromk+1. The probability function and factorial moments of these distributions are expressed in terms of the (right) truncated Bell (partition) polynomials and several special cases are briefly examined. Finally a Poisson process of orderk, leading in particular to the Poisson distribution of orderk, is discussed.