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

TL;DR

This paper introduces new factorizations of birth-death chains on integers using absorbing and reflecting chains, explores Darboux transformations, and connects these to spectral matrix transformations like Geronimus and Christoffel, with applications to chains with constant probabilities.

Contribution

It develops a novel factorization approach for birth-death chains on integers, including conditions for stochasticity and spectral transformations, expanding the understanding of chain transformations.

Findings

New absorbing-reflecting factorizations for birth-death chains.

Identification of spectral matrices as Geronimus and Christoffel transformations.

Application to chains with constant transition probabilities.

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.