Scalar-multi-tensorial equivalence for higher order $f\left(
  R,\nabla_{\mu}
  R,\nabla_{\mu_{1}}\nabla_{\mu_{2}}R,...,\nabla_{\mu_{1}}...\nabla_{\mu_{n}
  }R\right)$ theories of gravity
  The equivalence between theories depending on the derivatives of $R$, i.e.
$f\left( R,\nabla R,...,\nabla^{n}R\right) $, and scalar-multi-tensorial
theories is verified. The analysis is done in both metric and Palatini
formalisms. It is shown that $f\left( R,\nabla R,...,\nabla^{n}R\right) $
theories are equivalent to scalar-multi-tensorial ones resembling Brans-Dicke
theories with kinetic terms $\omega_{0}=0$ and $\omega_{0}= - \frac{3}{2}$ for
metric and Palatini formalisms respectively. This result is analogous to what
happens for $f(R)$ theories. It is worthy emphasizing that the
scalar-multi-tensorial theories obtained here differ from Brans-Dicke ones due
to the presence of multiple tensorial fields absent in the last. Furthermore,
sufficient conditions are established for $f\left( R,\nabla
R,...,\nabla^{n}R\right) $ theories to be written as scalar-multi-tensorial
theories. Finally, some examples are studied and the comparison of $f\left(
R,\nabla R,...,\nabla^{n}R\right) $ theories to $f\left( R,\Box
R,...\Box^{n}R\right) $ theories is performed.
