On asymptotic structure of continuous-time Markov branching processes allowing immigration without higher-order moments

We consider a continuous-time Markov branching process allowing immigration. Our main analytical tool is the slow variation (or more general, a regular variation) conception in the sense of Karamata. The slow variation property arises in many issues, but it usually remains rather hidden. For example, denoting by $p(n)$ the perimeter of an equilateral polygon with $n$ sides inscribed in a circle with a diameter of length $d$, one can check that the function $\boldsymbol{\pi}(n):={p(n)}/d$ converges to $\pi$ in the sense of Archimedes, but it slowly varies at infinity in the sense of Karamata. In fact, it is known that $p(n)=dn\sin{\left(\pi/n\right)}$ and then it follows $\boldsymbol{\pi}(\lambda{x}) /\boldsymbol{\pi}(x) \to 1$ as $x \to \infty$ for each $\lambda > 0$. Thus, $\boldsymbol{\pi}(x)$ is so slowly approaching $\pi$ that it can be suspected that "$\pi$ is not quite constant".
Application of Karamata functions in the branching processes theory allows one to bypass severe constraints concerning existence of the higher-order moments of the infinitesimal characteristics of the process under study. Zolotarev was one of the first who demonstrated an encouraging prospect of application of the slow variation conception in the theory of Markov branching processes and has obtained principally new results on asymptote of the survival probability of the process without immigration.
In this paper, delving deeply in the nature of the Karamata functions, we study more subtle properties of branching processes allowing immigration. In particular, under quite admissible conditions, we find explicit forms for the generating functions of invariant measures for the process under consideration.