A conjecture of Han on 3-cores and modular forms (Q486030)

Summary: In his study of Nekrasov-Okounkov type formulas on ``partition theoretic'' expressions for families of infinite products, Han discovered seemingly unrelated \(q\)-series that are supported on precisely the same terms as these infinite products. In collaboration with Ono, Han [\textit{G. N. Han} and \textit{K. Ono}, Ann. Comb. 15, No. 2, 305--312 (2011; Zbl 1233.05030)] proved one instance of this occurrence that exhibited a relation between the numbers \(a(n)\) that are given in terms of hook lengths of partitions, with the numbers \(b(n)\) that equal the number of 3-core partitions of \(n\). Recently Han revisited the \(q\)-series with coefficients \(a(n)\) and \(b(n)\), and numerically found a third \(q\)-series whose coefficients appear to be supported on the same terms. Here we prove Han's conjecture about this third series by proving a general theorem about this phenomenon.