  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.
