Grothendieck groups of repetitive cluster categories

In order to study cluster-tilted algebras and their intermediate coverings, Zhu introduced the notion of repetitive cluster categories, defined as the orbit categories ${\mathcal{D}}^b(\mathcal{H})\mathrm{/}⟨({\tau }^{-1}\mathrm{\Sigma }{)}^p⟩$ for $1\le p\in \mathbb{N}$, where $\mathcal{H}$ is a hereditary abelian category with tilting objects. In this paper, we compute partial but essential results on the Grothendieck groups of the repetitive cluster categories ${\mathcal{D}}^b(\mathrm{m}\mathrm{o}\mathrm{d}K{A}_n)\mathrm{/}⟨({\tau }^{-1}\mathrm{\Sigma }{)}^p⟩$ and ${\mathcal{D}}^b(\mathrm{m}\mathrm{o}\mathrm{d}K{D}_n)\mathrm{/}⟨({\tau }^{-1}\mathrm{\Sigma }{)}^p⟩$. Our results extend the known computations for classical cluster categories, reveal new structural patterns arising from the repetitive parameter $p$, and provide further evidence of the close interplay between Grothendieck groups, Auslander-Reiten theory, and Coxeter transformations.