  In Cuzinatto et al. [Phys. Rev. D 93, 124034 (2016)], it has been
demonstrated that theories of gravity in which the Lagrangian includes terms
depending on the scalar curvature $R$ and its derivatives up to order $n$, i.e.
$f\left(R,\nabla_{\mu}R,\nabla_{\mu_{1}}\nabla_{\mu_{2}}R,\dots,\nabla_{\mu_{1}}\dots\nabla_{\mu_{n}}R\right)$
theories of gravity, are equivalent to scalar-multitensorial theories in the
Jordan frame. In particular, in the metric and Palatini formalisms, this
scalar-multitensorial equivalent scenario shows a structure that resembles that
of the Brans-Dicke theories with a kinetic term for the scalar field with
$\omega_{0}=0$ or $\omega_{0}=-3/2$, respectively. In the present work, the
aforementioned analysis is extended to the Einstein frame. The conformal
transformation of the metric characterizing the transformation from Jordan's to
Einstein's frame is responsible for decoupling the scalar field from the scalar
curvature and also for introducing a usual kinetic term for the scalar field in
the metric formalism. In the Palatini approach, this kinetic term is absent in
the action. Concerning the other tensorial auxiliary fields, they appear in the
theory through a generalized potential. As an example, the analysis of an
extension of the Starobinsky model (with an extra term proportional to
$\nabla_{\mu}R\nabla^{\mu}R$) is performed and the fluid representation for the
energy-momentum tensor is considered. In the metric formalism, the presence of
the extra term causes the fluid to be an imperfect fluid with a heat flux
contribution; on the other hand, in the Palatini formalism the effective
energy-momentum tensor for the extended Starobinsky gravity is that of a
perfect fluid type. Finally, it is also shown that the extra term in the
Palatini formalism represents a dynamical field which is able to generate an
inflationary regime without a graceful exit.
