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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

New weighted Rogers-Ramanujan partition theorems and their implications


Authors: Krishnaswami Alladi and Alexander Berkovich
Journal: Trans. Amer. Math. Soc. 354 (2002), 2557-2577
MSC (2000): Primary 11P83, 11P81; Secondary 05A19
Published electronically: March 11, 2002
MathSciNet review: 1895193
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Abstract: This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of Göllnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at least two. Consequences of this include Jacobi's celebrated triple product identity for theta functions, Sylvester's famous refinement of Euler's theorem, as well as certain weighted partition identities. Next, by studying partitions with prescribed bounds on successive ranks and replacing these with weighted Rogers-Ramanujan partitions, we obtain two new sets of theorems - a set of three theorems involving partitions into parts $\not \equiv 0, \pm i$ (mod 6), and a set of three theorems involving partitions into parts $\not \equiv 0, \pm i$ (mod 7), $i=1,2,3$.


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Additional Information

Krishnaswami Alladi
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email: alladi@math.ufl.edu

Alexander Berkovich
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email: alexb@math.ufl.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-02-02977-X
PII: S 0002-9947(02)02977-X
Keywords: G\"{o}llnitz theorem, Rogers-Ramanujan partitions, method of weighted words, Jacobi triple product identity, Sylvester's theorem, weighted partition identities, successive ranks
Received by editor(s): September 1, 2001
Published electronically: March 11, 2002
Additional Notes: Research of the first author supported in part by the National Science Foundation Grant DMS 0088975
Research of the second author supported in part by a University of Florida CLAS Research Award
Article copyright: © Copyright 2002 American Mathematical Society