Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some theta function identities related to
the Rogers-Ramanujan continued fraction


Author: Seung Hwan Son
Journal: Proc. Amer. Math. Soc. 126 (1998), 2895-2902
MSC (1991): Primary 33D10; Secondary 11A55
MathSciNet review: 1458265
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In his first and second letters to Hardy, Ramanujan made several assertions about the Rogers-Ramanujan continued fraction $F(q)$. In order to prove some of these claims, G. N. Watson established two important theorems about $F(q)$ that he found in Ramanujan's notebooks. In his lost notebook, after stating a version of the quintuple product identity, Ramanujan offers three theta function identities, two of which contain as special cases the celebrated two theorems of Ramanujan proved by Watson. Using addition formulas, the quintuple product identity, and a new general product formula for theta functions, we prove these three identities of Ramanujan from his lost notebooks.


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

  • 1. Bruce C. Berndt, Ramanujan’s notebooks. Part III, Springer-Verlag, New York, 1991. MR 1117903 (92j:01069)
  • 2. Bruce C. Berndt, Ramanujan’s notebooks. Part IV, Springer-Verlag, New York, 1994. MR 1261634 (95e:11028)
  • 3. Bruce C. Berndt and Heng Huat Chan, Some values for the Rogers-Ramanujan continued fraction, Canad. J. Math. 47 (1995), no. 5, 897–914. MR 1350640 (97a:33043), http://dx.doi.org/10.4153/CJM-1995-046-5
  • 4. B. C. Berndt, H. H. Chan and L.-C. Zhang, Explicit evaluations of the Rogers-Ramanujan continued fraction, J. Reine Angew. Math. 480 (1996), 141-159. CMP 97:04
  • 5. B. C. Berndt, S.-S. Huang, J. Sohn and S. Son, Some theorems on the Rogers-Ramanujan continued fraction in Ramanujan's lost notebook (preprint).
  • 6. Bruce C. Berndt and Robert A. Rankin, Ramanujan, History of Mathematics, vol. 9, American Mathematical Society, Providence, RI; London Mathematical Society, London, 1995. Letters and commentary. MR 1353909 (97c:01034)
  • 7. Srinivasa Ramanujan, Notebooks. Vols. 1, 2, Tata Institute of Fundamental Research, Bombay, 1957. MR 0099904 (20 #6340)
  • 8. Srinivasa Ramanujan, The lost notebook and other unpublished papers, Springer-Verlag, Berlin; Narosa Publishing House, New Delhi, 1988. With an introduction by George E. Andrews. MR 947735 (89j:01078)
  • 9. L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318-343.
  • 10. G. N. Watson, Theorems stated by Ramanujan (VII): Theorems on continued fractions, J. London Math. Soc. 4 (1929), 39-48.
  • 11. E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469 (97k:01072)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 33D10, 11A55

Retrieve articles in all journals with MSC (1991): 33D10, 11A55


Additional Information

Seung Hwan Son
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green St., Urbana, Illinois 61801
Email: son@math.uiuc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04516-X
PII: S 0002-9939(98)04516-X
Keywords: Rogers-Ramanujan continued fraction, Euler's pentagonal number theorem, Jacobi triple product identity, quintuple product identity
Received by editor(s): February 21, 1997
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 1998 American Mathematical Society