On Ramanujan’s continued fraction for $(q^2;q^3)_{\infty }/(q;q^3)_{\infty }$
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- by George E. Andrews, Bruce C. Berndt, Jaebum Sohn, Ae Ja Yee and Alexandru Zaharescu PDF
- Trans. Amer. Math. Soc. 355 (2003), 2397-2411 Request permission
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
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Additional Information
- George E. Andrews
- Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
- MR Author ID: 26060
- Email: andrews@math.psu.edu
- Bruce C. Berndt
- Affiliation: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801
- MR Author ID: 35610
- 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
- MR Author ID: 186235
- Email: zaharesc@math.uiuc.edu
- 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. - © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1973995