Absorbing-reflecting factorizations for birth-death chains on the integers and their Darboux transformations

Abstract

We consider a new way of factorizing the transition probability matrix of a discrete-time birth-death chain on the integers by means of an absorbing and a reflecting birth-death chain to the state 0 and viceversa. First we will consider reflecting-absorbing factorizations of birth-death chains on the integers. We give conditions on the two free parameters such that each of the factors is a stochastic matrix. By inverting the order of the factors (also known as a Darboux transformation) we get new families of "almost" birth-death chains on the integers with the only difference that we have new probabilities going from the state $1$ to the state $-1$ and viceversa. On the other hand an absorbing-reflecting factorization of birth-death chains on the integers is only possible if both factors are splitted into two separated birth-death chains at the state $0$. Therefore it makes more sense to consider absorbing-reflecting factorizations of "almost" birth-death chains with extra transitions between the states $1$ and $-1$ and with some conditions. This factorization is now unique and by inverting the order of the factors we get a birth-death chain on the integers. In both cases we identify the spectral matrices associated with the Darboux transformation, the first one being a Geronimus transformation and the second one a Christoffel transformation of the original spectral matrix. We apply our results to examples of chains with constant transition probabilities.