New weighted RogersRamanujan partition theorems and their implications
Authors:
Krishnaswami Alladi and Alexander Berkovich
Journal:
Trans. Amer. Math. Soc. 354 (2002), 25572577
MSC (2000):
Primary 11P83, 11P81; Secondary 05A19
Published electronically:
March 11, 2002
MathSciNet review:
1895193
Fulltext PDF Free Access
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Abstract: This paper has a twofold purpose. First, by considering a reformulation of a deep theorem of Göllnitz, we obtain a new weighted partition identity involving the RogersRamanujan 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 RogersRamanujan 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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 K. Alladi, Partition identities involving gaps and weights, Trans. Amer. Math. Soc. 349 (1997), 50015019. MR 98c:05012
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 K. Alladi, Partition identities involving gaps and weights II, The Ramanujan J. 2 (1998), 2137. MR 2000a:11149
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 K. Alladi, On a partition theorem of Göllnitz and quartic transformations (with an appendix by B. Gordon), J. Number Theory 69 (1998), 153180. MR 99d:11112
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 K. Alladi, G. E. Andrews, and B. Gordon, Generalizations and refinements of a partition theorem of Göllnitz, Jour. Reine Angew. Math. 460 (1995), 165188. MR 96c:11119
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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/S000299470202977X
PII:
S 00029947(02)02977X
Keywords:
G\"{o}llnitz theorem,
RogersRamanujan 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
