Distribution of the sum of independent unity-truncated logarithmic series variables

Let X1, X2, ..., Xn be n independent and identically distributed random variables having the unity-truncated logarithmic series distribution with probability function given by f(x;θ) = αθX/x x ε T where α = [-log(1-θ) - θ]-1, 0 < θ < 1, and T = {2,3,...,∞}. Define their sum as Z = X1 + X2 + ... + Xn.We derive here the distribution of Z, denoted by p(z;n,θ), using the inversion formula for characteristic functions, in an explicit form in terms of a linear combination of Stirling numbers of the first kind. A recurrence relation for the probability function p(z;n,θ) is obtained and is utilized to provide a short table of p(z;n,θ) for certain values of n and θ. Furthermore, some properties of p(z;n,θ) are investigated following Patil and Wani [Sankhya, Series A, 27, (1965), 271-280].