The simple (linear) birth-and-death process is a widely used stochastic model for describing the dynamics of a population. When the process is observed discretely over time, despite the large amount of literature on the subject, little is known about formal estimator properties. Here we will show that its application to observed data is further complicated by the fact that numerical evaluation of the well-known transition probability is an ill-conditioned problem. To overcome this difficulty we will rewrite the transition probability in terms of a Gaussian hypergeometric function and subsequently obtain a three-term recurrence relation for its accurate evaluation. We will also study the properties of the hypergeometric function as a solution to the three-term recurrence relation. We will then provide formulas for the gradient and Hessian of the log-likelihood function and conclude the article by applying our methods for numerically computing maximum likelihood estimates in both simulated and real dataset.

简单的（线性的）生死过程是用于描述种群动态的广泛使用的随机模型。当随时间离散地观察该过程时，尽管有关该主题的文献很多，但对形式估计量的性质知之甚少。在这里，我们将表明，由于众所周知的跃迁概率的数值评估是一个病态问题，它在观测数据中的应用更加复杂。为了克服这个困难，我们将根据高斯超几何函数重写过渡概率，并随后获得三项递归关系以对其进行精确评估。我们还将研究超几何函数的性质，以解决三项递归关系。