Extended Sibuya distribution in the Subcritical Markov Branching processes

The subcritical Markov branching process X(t) starting with one particle as the initial condition has the ultimate extinction probability q = 1. The branching mechanism in consideration is defined by the mixture of logarithmic distributions on the nonnegative integers. The purpose of the present paper is to prove that in this case the random number of particles X(t) alive at time t>0 follows the shifted extended Sibuya distribution, with parameters depending on the time $t>0$. The conditional limit probability is exactly the logarithmic series distribution supported by the positive integers.