The third Kolmogorov equation for a branching process with interaction of particles. (Q1432226)

Let \(\mu (t),\, t\in [0,\infty)\), be a continuous time homogeneous Markov process with a countable number of states \(N=\{0,1,2,\ldots\}\) and infinitesimal transition probabilities \(a_{ij}\) equal to \(i(i-1)p_{j-i+2}\) if \(j\geq i-2,\;i\neq j,\) equal to \( -i(i-1) \) if \(i=j\) and equal to \(0\) if \(j<i-2,\) where \(p_k\geq 0\), \(\sum_{k=0}^\infty p_k=1\), \(p_2=0.\) Let \(P_{ij}(t)=P(\mu (t)=j\mid \mu (0)=i)\) be the transition probabilities of this process. Set \(h(s)=\sum_{k=0}^\infty p_ks^k\) and \[ {\mathcal F}(t;z,s)=\sum_{i=0}^\infty \sum_{j=0}^\infty\frac{z^i}{i!}P_{ij}(t)s^j. \] The author shows that \[ (h(s)-s^{2})\frac{\partial ^{2}\mathcal{F}}{\partial s^{2}}-z^{2}\left(h \left(\frac{ \partial }{\partial z}\right)- \frac{\partial ^{2}}{\partial z^{2}} \right)\mathcal{F}=0 \] and calls the obtained equation as the third Kolmogorov equation for a branching process with interaction of particles. For the case \(h(s)=s^2\) an explicit solution of the equation above is given.