New Identities for 7-cores with prescribed BG-rank

Details New Identities for 7-cores with prescribed BG-rank New Identities for 7-cores with prescribed BG-rank

2006-03-06

arxiv.org/abs/math/0603150v2

Mathematics

Number Theory

12 pages

Scientific paper

A q-series with nonnegative power series coefficients is called positive. The partition statistics BG-rank is defined as an alternating sum of parities of parts of a partition. It is known that the generating function for the number of partitions of n that are 7-cores with given BG-rank can be written as certain sum of multi-theta functions. We give explicit representations for these generating functions in terms of sums of positive eta-quotients and derive inequalities for the their coefficients. New identities for the generating function of unrestricted 7-cores and inequalities for their coefficients are also obtained. Our proofs utilize Ramanujan's theory of modular equations.

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