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A birth-death process is a continuous-time Markov chain that counts thenumber of particles in a system over time. In the general process with $n$current particles, a new particle is born with instantaneous rate $\lambda_n$and a particle dies with instantaneous rate $\mu_n$. Currently no robust andefficient method exists to evaluate the finite-time transition probabilities ina general birth-death process with arbitrary birth and death rates. In thispaper, we first revisit the theory of continued fractions to obtain expressionsfor the Laplace transforms of these transition probabilities and make explicitan important derivation connecting transition probabilities and continuedfractions. We then develop an efficient algorithm for computing theseprobabilities that analyzes the error associated with approximations in themethod. We demonstrate that this error-controlled method agrees with knownsolutions and outperforms previous approaches to computing these probabilities.Finally, we apply our novel method to several important problems in ecology,evolution, and genetics.

Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution