On Ramanujan's continued fraction for

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 -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.

**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****11.**S. Ramanujan,*Notebooks*(2 volumes), Tata Institute of Fundamental Research, Bombay, 1957. MR**20:6340****12.**S. Ramanujan,*The Lost Notebook and Other Unpublished Papers*, Narosa, New Delhi, 1988. MR**89j:01078****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:
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