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ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Author(s): George E. Andrews; Bruce C. Berndt; Jaebum Sohn; Ae Ja Yee; Alexandru Zaharescu
Journal: Trans. Amer. Math. Soc. 355 (2003), 2397-2411.
MSC (2000): Primary 33Dxx; Secondary 11B65, 11A55, 30B70
Posted: September 25, 2002
MathSciNet review: 1973995
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Abstract: The continued fraction in the title is perhaps the deepest of Ramanujan's -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:

1.: G. E. Andrews, On -difference equations for certain well-poised basic hypergeometric series, Quart. J. Math. (Oxford) 19 (1968), 433-447. MR 38:6112
2.: G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, MA, 1976. MR 58:27738
3.: G. E. Andrews, An introduction to Ramanujan's ``lost'' notebook, Amer. Math. Monthly 86 (1979), 89-108. MR 80e:01018
4.: G. E. Andrews, A page from Ramanujan's lost notebook, Indian J. Math. 32 (1990), 207-216. MR 92b:11072
5.: G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part I, Springer-Verlag, New York, to appear.
6.: G. E. Andrews, B. C. Berndt, L. Jacobsen, and R. L. Lamphere, The Continued Fractions Found in the Unorganized Portions of Ramanujan's Notebooks, Memoirs Amer. Math. Soc. No. 477, 99 (1992). MR 93f:11008
7.: G. E. Andrews, B. C. Berndt, J. Sohn, A. J. Yee, and A. Zaharescu, Continued fractions with three limit points, submitted for publication.
8.: B. C. Berndt, Ramanujan's Notebooks, Part V, Springer-Verlag, New York, 1998. MR 99f:11024
9.: B. C. Berndt and J. Sohn, Asymptotic formulas for two continued fractions in Ramanujan's lost notebook, J. London Math. Soc. (2) 65 (2002), 271-284.
10.: L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North Holland, Amsterdam, 1992. MR 93g:30007
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13.: A. Selberg, Über einige arithmetische Identitäten Avh. Norske Vid.-Akad. Oslo I. Mat.-Naturv. Kl, No. 8, (1936), 3-23.

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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: 10.1090/S0002-9947-02-03155-0
PII: S 0002-9947(02)03155-0
Received by editor(s): October 5, 2001
Posted: 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 of article: Copyright 2002, American Mathematical Society

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