Classification of the ring of Witt vectors and the necklace ring associated with the formal group law X+Y−qXY

In this paper, we classify the ring of Witt vectors and the necklace ring associated with the formal group law X+Y−qXY up to strict-isomorphism as q varies over the set of integers. We also introduce a q-deformation of the Grothendieck ring of formal power series with constant term 1 over rings in which q⋅1 is a unit. Various bijections related to Witt vectors, necklaces, and formal power series are given. As an application, we provide a q-analogue of the celebrated cyclotomic identity and its bijective proof.