An Asymptotic Expansion of a 2F1 Function with Applications
Abstract
The 2F1 hypergeometric function appears in a wide variety of appli- cations in probability and statistics. This article is motivated from an important application that arises in stochastic processes. A birth and death process that starts with a single species and birth rate larger than the death rate may lead to trees of species or genera. Under standard assumptions on the process, the probability of the size of a genera is a discrete distribution that can easily be computed at time t from the ori- gin. However, when the process leads to several sublineages of trees that originated at dierent times from the origin, then the distribution of the size of the population commonly referred to as taxon size distribution of the genus size must be weighted by the dierent times that the genera originate. It was shown in Moschopoulos and Shpak (2010), that under the exponential distribution for time t the taxon distribution may be ex- pressed as an 2F1(x; b; x + b; ) hypergeometric function. This form has not been exploited in the literature. In this article we consider an asymp- totic expansion of this 2F1 for large x in terms of generalized Bernoulli polynomials that can be computed up to any degree of accuracy.