Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the quintuple product identity

Authors: Hershel M. Farkas and Irwin Kra
Journal: Proc. Amer. Math. Soc. 127 (1999), 771-778
MSC (1991): Primary 30F30, 11F03; Secondary 30B99, 14H05, 05A30
MathSciNet review: 1487364
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this note we present a new proof of the quintuple product identity which is based on our study of $ k^{th} $ order theta functions with characteristics and the identities they satisfy. In this context the quintuple product identity is another example of an identity which when phrased in terms of theta functions, rather than infinite products and sums, has a simpler form and is much less mysterious.

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

  • 1. K. Alladi, The quintuple product identity and shifted partition functions, Comput. Math. Appl. 68 (1996), 3-13. MR 98c:05012
  • 2. R. Brooks, H.M. Farkas, and I. Kra, Number theory, theta identities and modular curves, Contemporary Math. 201 (1997), 125-154. CMP 97:07
  • 3. L. Carlitz and M.V. Subbarao, A simple proof of the quintuple product identity, Proc. Amer. Math. Soc. 32 (1972), 42-44. MR 44:6507
  • 4. H.M. Farkas and Y. Kopeliovich, New theta constant identities II, Proc. Amer. Math. Soc. 123 (1995), 1009-1020. MR 95e:11050
  • 5. H.M. Farkas, Y. Kopeliovich, and I. Kra, Uniformization of modular curves, Comm. Anal. Geom. 4 (1996), 207-259. MR 97j:11019a
  • 6. H.M. Farkas and I. Kra, Theta constants, Riemann surfaces and the modular group, in preparation.
  • 7. -, A function theoretic approach to the Ramanujan partition identities with applications to combinatorial number theory, Proc. of Iberoamerican Congress on Geometry, Chile 1998, pp. 75-106.
  • 8. B. Gordon, Some identities in combinatorial analysis, Quart. J. Math. Oxford 12 (1961), 285-290. MR 25:21
  • 9. G.N. Watson, Theorems stated by Ramanujan, (VII): Theorems on continued fractions, J. London Math.Soc. 4 (1929), 39-48.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30F30, 11F03, 30B99, 14H05, 05A30

Retrieve articles in all journals with MSC (1991): 30F30, 11F03, 30B99, 14H05, 05A30

Additional Information

Hershel M. Farkas
Affiliation: Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

Irwin Kra
Affiliation: Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794

Keywords: $q$-series, $k^{th}$ order theta functions with characteristics, partitions of integers, Jacobi triple product, Euler pentagonal number theorem
Received by editor(s): June 17, 1997
Additional Notes: The second author’s research was supported in part by NSF Grant DMS 9500557. The first author’s research was supported in part by the Gabriella and Paul Rosenbaum Foundation and the Edmund Landau Center for Research in Mathematical Analysis sponsored by the Minerva Foundation Germany. Both authors were supported in part by a US-Israel BSF Grant 95-348.
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 1999 American Mathematical Society