Birth-death processes and 𝑞-continued fractions

Feng, Tony; Kirsch, Rachel; Villella, Elise; Wage, Matt

doi:10.1090/S0002-9947-2012-05522-X

Birth-death processes and $q$-continued fractions
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by Tony Feng, Rachel Kirsch, Elise Villella and Matt Wage PDF

Trans. Amer. Math. Soc. 364 (2012), 2703-2721 Request permission

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.

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