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 $\rm{Ein} (z)$.

翻译：特殊函数在物理学和数学中始终扮演着核心角色，它们作为非线性微分方程的解出现，同时也广泛应用于分支过程理论中，该理论大量使用概率生成函数。实函数迭代理论导出了离散时间与实时马尔可夫分支过程的极限定理。通过积分柯尔莫哥洛夫方程，分析了实时情况下的粒子泊松繁殖过程。这些结果通过采用亚临界与临界分支机制下柯尼希函数的图形表示得到了进一步扩展。文中还讨论了亚临界情形下的极限条件定律以及临界情形下的不变测度。所获得的显式解包含了指数贝尔多项式和修正的指数积分函数 $\rm{Ein} (z)$。