Field equations and Noether potentials for higher-order theories of gravity with Lagrangians involving $\Box^i R$, $\Box^i R_{\mu\nu}$ and $\Box^i R_{\mu\nu\rho\sigma}$

In this paper, we aim to perform a systematical investigation on the field equations and Noether potentials for the higher-order gravity theories endowed with Lagrangians depending on the metric and the Riemann curvature tensor, together with $i$th ($i=1,2,\cdot\cdot\cdot$) powers of the Beltrami-d'Alembertian operator $\Box$ acting on the latter. We start with a detailed derivation of the field equations and the Noether potential corresponding to the Lagrangian $\sqrt{-g}L_R(R,\Box R,\cdot\cdot\cdot,\Box^m R)$ through the direct variation of the Lagrangian and a method based upon the conserved current. Next the parallel analysis is extended to a more generic Lagrangian $\sqrt{-g}L_{\text{Ric}}(g^{\mu\nu}, R_{\mu\nu},\Box R_{\mu\nu}, \cdot\cdot\cdot,\Box^m R_{\mu\nu})$, as well as to the generalization of the Lagrangian $\sqrt{-g}L_{\text{Ric}}$, which depends on the metric $g^{\mu\nu}$, the Riemann tensor $R_{\mu\nu\rho\sigma}$ and $\Box^i R_{\mu\nu\rho\sigma}$s. Finally, all the results associated to the three types of Lagrangians are extended to the Lagrangian relying on an arbitrary tensor and the variables via $\Box^i$ acting on such a tensor. In particular, we take into consideration of equations of motion and Noether potentials for nonlocal gravity models. For Lagrangians involving the variables $\Box^i R$, $\Box^i R_{\mu\nu}$ and $\Box^i R_{\mu\nu\rho\sigma}$, our investigation provides their concrete Noether potentials and the field equations without the derivative of the Lagrangian density with respect to the metric. Besides, the Iyer-Wald potentials associated to these Lagrangians are also presented.