Markov branching random walks on $\mathbf{Z}_+$. Approach using orthogonal polynomials

We consider a continuous-time homogeneous Markov process on the state space $\mathbf{Z}_+=\{0,1,2,\dots\}$. The process is interpreted as the motion of a particle. A particle may transit only to neighboring points $\mathbf{Z}_+$, i.e., for each single motion of the particle, its coordinate changes by 1. The process is equipped with a branching mechanism. Branching sources may be located at each point of $\mathbf{Z}_+$. At a moment of branching, new particles appear at the branching point and then evolve independently of each other (and of the other particles) by the same rules as the initial particle. To such a branching Markov process there corresponds a Jacobi matrix. In terms of orthogonal polynomials corresponding to this matrix, we obtain formulas for the mean number of particles at an arbitrary fixed point of $\mathbf{Z}_+$ at time $t>0$. The results obtained are applied to some concrete models, an exact value for the mean number of particles in terms of special functions is given, and an asymptotic formula for this quantity for large time is found.