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We consider discrete-time birth-death chains on a spider, i.e. a graphconsisting of $N$ discrete half lines on the plane that are joined at theorigin. This process can be identified with a discrete-time quasi-birth-deathprocess on the state space $\mathbb{N}_0 \times \{1, 2, \dots, N\},$represented by a block tridiagonal transition probability matrix. We prove thatwe can analyze this process by using spectral methods and obtain the $n$-steptransition probabilities in terms of a weight matrix and the correspondingmatrix-valued orthogonal polynomials (the so-called Karlin-McGregor formula).We also study under what conditions we can get a reflecting-absorbingfactorization of the birth-death chain on a spider which can be seen as astochastic UL block factorization of the transition probability matrix of thequasi-birth-death process. With this factorization we can perform a discreteDarboux transformation and get new families of "almost" birth-death chains on aspider. The spectral matrix associated with the Darboux transformation will bea Geronimus transformation of the original spectral matrix. Finally, we applyour results to the random walk on a spider, i.e. with constant transitionprobabilities.

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