Birth-death chains on a spider: Spectral analysis and reflecting-absorbing factorization

We consider discrete-time birth-death chains on a spider, i.e. graph consisting of N discrete half lines the plane that are joined at origin. This process can be identified with quasi-birth-death state space N0×{1,2,…,N}, represented by block tridiagonal transition probability matrix. prove we analyze this using spectral methods and obtain n-step probabilities in terms weight matrix corresponding matrix-valued orthogonal polynomials (the so-called Karlin-McGregor formula). also study under what conditions get reflecting-absorbing factorization chain spider which seen as stochastic UL process. With perform Darboux transformation new families “almost” spider. The associated will Geronimus original Finally, apply our results to random walk constant probabilities.