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We obtain effective asymptotics for the coefficients of q-expansions which involve partial theta functions. As applications, we obtain effective asymptotics for ranks, and cranks of integer partitions. Moreover, from these effective estimates, we obtain asymptotic inequalities among them with effective lower bounds.
Center for Applications of Mathematical Principles, National Institute for Mathematical Sciences, 70 Yuseong-daero 1689-gil, Yuseong-gu, Daejeon 305-811, Republic of Korea