Birth-death processes and $q$-continued fractions

Abstract

Abstract. In the 1997 paper of Parthasarathy, Lenin, Schoutens, and Van Assche, the authors study a birth-death process related to the Rogers-Ramanujan continued fraction r(q). We generalize their results to establish a correspondence between birth-death processes and a larger family of q-continued fractions. It turns out that many of these continued fractions, including r(q), play important roles in number theory, specifically in the theory of modular forms and q-series. We draw upon the number-theoretic properties of modular forms to give identities between the transition probabilities of different birth-death processes.