Koenigs functions in the subcritical and critical Markov branching processes with Poisson probability reproduction of particles

Special functions have always played a central role in physics and in mathematics, arising as solutions of nonlinear differential equations, as well as in the theory of branching processes, which extensively uses probability generating functions. The theory of iteration of real functions leads to limit theorems for the discrete-time and real-time Markov branching processes. The Poisson reproduction of particles in real time is analysed through the integration of the Kolmogorov equation. These results are further extended by employing graphical representations of Koenigs functions under subcritical and critical branching mechanisms. The limit conditional law in the subcritical case and the invariant measure for the critical case are discussed, as well. The obtained explicit solutions contain the exponential Bell polynomials and the modified exponential-integral function $\rm{Ein} (z)$.