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，它们既作为非线性微分方程的解出现，也广泛应用于分支过程理论中，后者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，分析了实时情形下粒子的泊松繁殖过程。这些结果进一步借助亚临界与临界分支机制下柯尼希斯函数的图形表示得以拓展。文中亦讨论了亚临界情形下的极限条件律以及临界情形下的不变测度。所获得的显式解包含指数贝尔多项式与修正的指数积分函数 $\rm{Ein} (z)$。