On Birth-Death Processes with Rational Growth Rates

Abstract

Birth-death processes can be described by systems of differential equations together with certain probability conditions. Karlin and McGregor have shown that these systems of equations and the probability conditions have solutions which can be expressed by an integral representation formula which uses an associated set of orthogonal polynomials and the distribution function for these polynomials. It is shown that certain general results from the study of orthogonal polynomials can be used to describe this distribution function in most cases where the birth and death rates are rational functions of the population size.