Jacobi branching random walks corresponding to orthogonal polynomials of discrete variable

A branching random walk on $\mathbf{Z}_+$ is considered, which corresponds to a Jacobi matrix. Previously, formulas for the average number of particles at an arbitrary fixed point in $\mathbf{Z}_+$ at time $t>0$ were obtained in terms of the orthogonal polynomials associated with this matrix. In the present work, the application of the obtained results to certain models involving orthogonal polynomials of a discrete variable (Krawtchouk, Meixner, and Poisson–Charlier polynomials) is discussed.