  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.
