${\varphi }_{ϵ}$-coordinated modules for vertex algebras

We study ${\varphi }_{ϵ}$-coordinated modules for vertex algebras, where ${\varphi }_{ϵ}$ with $ϵ$ an integer parameter is a family of associates of the one-dimensional additive formal group. As the main results, we obtain a Jacobi type identity and a commutator formula for ${\varphi }_{ϵ}$-coordinated modules. We then use these results to study ${\varphi }_{ϵ}$-coordinated modules for vertex algebras associated to Novikov algebras by Primc.