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On Ramanujan's continued fraction for $(q^2;q^3)_{\infty}/(q;q^3)_{\infty}$


Authors: George E. Andrews, Bruce C. Berndt, Jaebum Sohn, Ae Ja Yee and Alexandru Zaharescu
Journal: Trans. Amer. Math. Soc. 355 (2003), 2397-2411
MSC (2000): Primary 33Dxx; Secondary 11B65, 11A55, 30B70
DOI: https://doi.org/10.1090/S0002-9947-02-03155-0
Published electronically: September 25, 2002
MathSciNet review: 1973995
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Abstract: The continued fraction in the title is perhaps the deepest of Ramanujan's $q$-continued fractions. We give a new proof of this continued fraction, more elementary and shorter than the only known proof by Andrews, Berndt, Jacobsen, and Lamphere. On page 45 in his lost notebook, Ramanujan states an asymptotic formula for a continued fraction generalizing that in the title. The second main goal of this paper is to prove this asymptotic formula.


References [Enhancements On Off] (What's this?)

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Additional Information

George E. Andrews
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: andrews@math.psu.edu

Bruce C. Berndt
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801
Email: berndt@math.uiuc.edu

Jaebum Sohn
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Email: sohn@math.ohio-state.edu

Ae Ja Yee
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801
Email: yee@math.uiuc.edu

Alexandru Zaharescu
Affiliation: Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest RO-70700, Romania
Address at time of publication: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801
Email: zaharesc@math.uiuc.edu

DOI: https://doi.org/10.1090/S0002-9947-02-03155-0
Received by editor(s): October 5, 2001
Published electronically: September 25, 2002
Additional Notes: The first author was supported in part by grant DMS-9206993 from the National Science Foundation.
The second author was supported in part by grant MDA904-00-1-0015 from the National Security Agency.
The fourth author was supported in part by the postdoctoral fellowship program from the Korea Science and Engineering Foundation, and by a grant from the Number Theory Foundation.
Article copyright: © Copyright 2002 American Mathematical Society

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