A family of discrete distributions

Ambagaspitiya and Balakrishnan (1994a) used the identity pn(a,b) = aa + bb + anpn - 1(a + b,b), n = 1,2,…, 0(a, b) = exp(−a) to obtain a recursive formula for computing the distribution function of the compound generalized Poisson distribution. In this paper, we consider the discrete distribution family with the property pn(a,b) = h1(a,b) + h2(a,b)npn - 1(a + b,b), n = 1,2,…, is a generalization of the first identity. We prove that weighted generalized Poisson distributions and weighted generalized negative binomial distributions with weights of the form w(a + bn; b) are two subclasses in the family. We provide a recursive formula for computation of respective compound distributions. Also we discuss the stability of the recursion as well as handling overflow / underflow problems.