New weighted Rogers-Ramanujan partition theorems and their implications

Alladi, Krishnaswami; Berkovich, Alexander

doi:10.1090/S0002-9947-02-02977-X

New weighted Rogers-Ramanujan partition theorems and their implications
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Trans. Amer. Math. Soc. 354 (2002), 2557-2577 Request permission

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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