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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
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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. B. C. Berndt, Ramanujan's Notebooks, Part III, Springer-Verlag, New York, 1991. MR 92j:01069
  • 2. B. C. Berndt, Ramanujan's Notebooks, Part IV, Springer-Verlag, New York, 1994. MR 95e:11028
  • 3. B. C. Berndt and H. H. Chan, Some values for the Rogers-Ramanujan continued fraction, Canadian J. Math. 47 (1995), 897-914. MR 97a:33043
  • 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. B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary, Amer. Math. Soc., Providence, 1995. MR 97c:01034
  • 7. S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957. MR 20:6340
  • 8. S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988. MR 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, 4 ed., University Press, Cambridge, 1996. MR 97k:01072

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Additional Information

Seung Hwan Son
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green St., Urbana, Illinois 61801

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

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