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

DOI:
https://doi.org/10.1090/S0002-9939-99-04791-7

MathSciNet review:
1487364

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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 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.

**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.

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Additional Information

**Hershel M. Farkas**

Affiliation:
Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

Email:
farkas@math.huji.ac.il

**Irwin Kra**

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

Email:
irwin@math.sunysb.edu

DOI:
https://doi.org/10.1090/S0002-9939-99-04791-7

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