Markov branching random walks on $\mathbf{Z}_+$ with absorption at zero

We consider a homogeneous Markov process with continuous time on $\mathbf{Z}_+=\{0,1,2,\dots \}$, which we interpret as the motion of a particle. A particle can only move to neighboring points $\mathbf{Z}_+$, that is, each time the particle's position changes, its coordinate changes by one. The process is equipped with a branching mechanism. Branch sources can be located at each point of $\mathbf{Z}_+$. At the moment of branching, new particles appear at the branch point and then begin to evolve independently of each other (and of other particles) according to the same laws as the initial particle. Point zero on the lattice $\mathbf{Z}_+$ is an absorbing state, that is, a particle with a non-zero probability can go to zero, but it instantly dies there. Such a branching random walk is associated with the Jacobian matrix. In terms of orthogonal polynomials of the second kind corresponding to the matrix, formulas are obtained for the average number of particles at an arbitrary fixed point of $\mathbf{Z}_+\setminus\{0\}$ at time $t>0$. The results are applied to some specific models, an exact value for the average number of particles is obtained in terms of special functions, and its asymptotic behavior is found at large times.