Some theta function identities related to the RogersRamanujan continued fraction
Author:
Seung Hwan Son
Journal:
Proc. Amer. Math. Soc. 126 (1998), 28952902
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 RogersRamanujan continued fraction . In order to prove some of these claims, G. N. Watson established two important theorems about 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.
 1.
Bruce
C. Berndt, Ramanujan’s notebooks. Part III,
SpringerVerlag, New York, 1991. MR 1117903
(92j:01069)
 2.
Bruce
C. Berndt, Ramanujan’s notebooks. Part IV,
SpringerVerlag, New York, 1994. MR 1261634
(95e:11028)
 3.
Bruce
C. Berndt and Heng
Huat Chan, Some values for the RogersRamanujan continued
fraction, Canad. J. Math. 47 (1995), no. 5,
897–914. MR 1350640
(97a:33043), http://dx.doi.org/10.4153/CJM19950465
 4.
B. C. Berndt, H. H. Chan and L.C. Zhang, Explicit evaluations of the RogersRamanujan continued fraction, J. Reine Angew. Math. 480 (1996), 141159. CMP 97:04
 5.
B. C. Berndt, S.S. Huang, J. Sohn and S. Son, Some theorems on the RogersRamanujan 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Ramanujan, The lost notebook and other unpublished papers,
SpringerVerlag, 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), 318343.
 10.
G. N. Watson, Theorems stated by Ramanujan (VII): Theorems on continued fractions, J. London Math. Soc. 4 (1929), 3948.
 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)
 1.
 B. C. Berndt, Ramanujan's Notebooks, Part III, SpringerVerlag, New York, 1991. MR 92j:01069
 2.
 B. C. Berndt, Ramanujan's Notebooks, Part IV, SpringerVerlag, New York, 1994. MR 95e:11028
 3.
 B. C. Berndt and H. H. Chan, Some values for the RogersRamanujan continued fraction, Canadian J. Math. 47 (1995), 897914. MR 97a:33043
 4.
 B. C. Berndt, H. H. Chan and L.C. Zhang, Explicit evaluations of the RogersRamanujan continued fraction, J. Reine Angew. Math. 480 (1996), 141159. CMP 97:04
 5.
 B. C. Berndt, S.S. Huang, J. Sohn and S. Son, Some theorems on the RogersRamanujan 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), 318343.
 10.
 G. N. Watson, Theorems stated by Ramanujan (VII): Theorems on continued fractions, J. London Math. Soc. 4 (1929), 3948.
 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 UrbanaChampaign, 1409 W. Green St., Urbana, Illinois 61801
Email:
son@math.uiuc.edu
DOI:
http://dx.doi.org/10.1090/S000299399804516X
PII:
S 00029939(98)04516X
Keywords:
RogersRamanujan 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
