New Weighted Rogers-Ramanujan Partition Theorems and their Implications

Details New Weighted Rogers-Ramanujan Partition Theorems and their Implications New Weighted Rogers-Ramanujan Partition Theorems and their Implications

2000-09-18

arxiv.org/abs/math/0009171v1

Mathematics

Combinatorics

31 pages, 4 figures

Scientific paper

This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of G\"ollnitz, 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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