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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Birth-death processes and $ q$-continued fractions


Authors: Tony Feng, Rachel Kirsch, Elise Villella and Matt Wage
Journal: Trans. Amer. Math. Soc. 364 (2012), 2703-2721
MSC (2010): Primary 03B48; Secondary 11A55
Published electronically: January 17, 2012
MathSciNet review: 2888225
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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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Additional Information

Tony Feng
Affiliation: 58 Plympton Street, 479 Quincy Mail Center, Cambridge, Massachusetts 02138
Email: tfeng@college.harvard.edu

Rachel Kirsch
Affiliation: 7212 Longwood Drive, Bethesda, Maryland 20817
Email: rkirsch@math.unl.edu

Elise Villella
Affiliation: 146 Harrison Drive, Edinboro, Pennsylvania 16412
Email: elisemccall@gmail.com

Matt Wage
Affiliation: 1411 N. Briarcliff Drive, Appleton, Wisconsin 54915
Email: mwage@princeton.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05522-X
PII: S 0002-9947(2012)05522-X
Received by editor(s): July 24, 2009
Received by editor(s) in revised form: October 26, 2010
Published electronically: January 17, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.