Partition implications of a three-parameter \(q\)-series identity (Q2181585)

The authors generalize a $q$-series identity found in the unorganized portion of Ramanujan's second and third notebooks. As an immediate application, they derive a three-parameter identity. Utilizing this identity, the authors not only generalize a recent result of \textit{G. E. Andrews} et al. [Acta Arith. 158, No. 3, 199--218 (2013; Zbl 1268.05019)], but obtain several new weighted partition identities and a natural proof of a recent result of \textit{F. G. Garvan} [Contemp. Math. Appl., Monogr. Expo. Lect. Notes 1, 239--249 (2018; Zbl 1456.11199)]. Moreover, the authors obtain a new identity consisting of an infinite series involving a special case of Fine's function $F(a, b; t)$, one special case of which yields Andrews' famous identity for the smallest parts function. This identity also unravels new relations between the divisor function and other partition-theoretic functions.