For a hereditary, finite dimensional k-algebra A the Coxeter transformation is a linear endomorphism ΦA:K0(modA)→K0(modA) of the Grothendieck group K0(modA) of modA such that ΦA[M]=[τAM] for every non-projective indecomposable A-module M, that is the Auslander-Reiten translation τA induces a linear endomorphism of K0(modA). It is natural to ask whether there are other algebras A having a linear endomorphism ΦA∈EndZ(K0(modA)) with ΦA[M]=[τAM] for all non-projective M∈indA. We will show that this is the case for all Nakayama algebras. Conversely, we will show that if an algebra A=kQ/I, where Q is a non-acyclic quiver and I⊲kQ is an admissible ideal, admits such a linear endomorphism then A is already a cyclic Nakayama algebra.