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 split 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 also study the probabilistic implications of both transformations. Finally, we apply our results to examples of chains with constant transition probabilities.

我们考虑一种新方法，通过吸收和反映状态0到状态反之亦然的生死链，来分解整数上离散时间生死链的转移概率矩阵。首先，我们将考虑在整数上反映出生-死亡链的吸收吸收因式分解。我们给出两个自由参数的条件，以使每个因子都是一个随机矩阵。通过反转因子的顺序（也称为Darboux变换），我们获得了整数上“几乎”出生-死亡链的新族，唯一的不同是，我们有从状态1到状态的新概率$-\mathrm{1个}$反之亦然。另一方面，只有当两个因子在状态0下都被分解成两个分离的生死链时，整数上的生死链的吸收反射分解才有可能。因此，考虑“几乎是“生死链”，在州1和州之间有额外的过渡$-\mathrm{1个}$并有一些条件。现在，这种分解是唯一的，并且通过反转因子的顺序，可以得到整数上的生死链。在这两种情况下，我们都确定了与Darboux变换相关的光谱矩阵，第一个是Geronimus变换，第二个是原始光谱矩阵的Christoffel变换。我们还研究了两种转换的概率含义。最后，我们将我们的结果应用于具有恒定转移概率的链的示例。