Periodic derived Hall algebras of hereditary abelian categories

Let m be a positive integer and $D_m\left(A\right)$ be the m-periodic derived category of a finitary hereditary abelian category $A$. Applying the derived Hall numbers of the bounded derived category $D^b\left(A\right)$, we define an m-periodic extended derived Hall algebra for $D_m\left(A\right)$, and use it to give a global, unified and explicit characterization for the algebra structure of Bridgeland's Hall algebra of periodic complexes. Moreover, we also provide an explicit characterization for the odd periodic derived Hall algebra of $A$ defined by Xu-Chen [24].