Scalar-multi-tensorial equivalence for higher order $f(R,{\mathrm{\nabla }}_{\mu }R,{\mathrm{\nabla }}_{{\mu }_1}{\mathrm{\nabla }}_{{\mu }_2}R,\mathrm{.}\mathrm{.}\mathrm{.},{\mathrm{\nabla }}_{{\mu }_1}\mathrm{.}\mathrm{.}\mathrm{.}{\mathrm{\nabla }}_{{\mu }_n}R)$ theories of gravity

The equivalence between theories depending on the derivatives of $R$, i.e. $f(R,\mathrm{\nabla }R,\mathrm{.}\mathrm{.}\mathrm{.},{\mathrm{\nabla }}^{n}R)$, and scalar-multi-tensorial theories is verified. The analysis is done in both metric and Palatini formalisms. It is shown that $f(R,\mathrm{\nabla }R,\mathrm{.}\mathrm{.}\mathrm{.},{\mathrm{\nabla }}^{n}R)$ 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) $theoriesto$f\left( R,\Box R,…\Box^{n}R\right) $ theories is performed.