On a modified binomial distribution of order k (Q804066)

The paper provides two extensions of the binomial distribution of order k defined by \textit{K. Hirano} [Fibonacci numbers and their applications, Pap. 1st Int. Conf., Patras/Greece 1984, Math. Appl., D. Reidel Publ. Co. 28, 43-53 (1986; Zbl 0601.62023)], derives their probability generating functions, moment generating functions and shows that the Poisson distribution is a limiting form of one of them. These are based on a generalization of the notion of success so as to be the occurrence of the event \[ FFF...F\quad SS...S\quad (F\quad k_ 1-fold\text{ and } S\quad k_ 2-fold) \] in a series of Bernoulli trials where S stands for a successful trial and F for an unsuccessful trial. Further it examines the distribution of the number of trials between two consecutive ``successes'' in the above sense.