Scalar-multi-tensorial equivalence for higher order $fleft( R, abla_{mu} R, abla_{mu_{1}} abla_{mu_{2}}R,..., abla_{mu_{1}}... abla_{mu_{n} }Rright)$ theories of gravity

The equivalence between theories depending on the derivatives of $R$, i.e. $fleft( R, abla R,..., abla^{n}Rright) $, and scalar-multi-tensorial theories is verified. The analysis is done in both metric and Palatini formalisms. It is shown that $fleft( R, abla R,..., abla^{n}Rright) $ 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 $fleft( R, abla R,..., abla^{n}Rright) $ theories to be written as scalar-multi-tensorial theories. Finally, some examples are studied and the comparison of $fleft( R, abla R,..., abla^{n}Rright) $ theories to $fleft( R,Box R,...Box^{n}Rright) $ theories is performed.