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 (mod 6), and a set of three theorems involving partitions into parts (mod 7), .

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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

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