  In this paper, we introduce two new generalized inverses of matrices, namely,
the $\bra{i}{m}$-core inverse and the $\pare{j}{m}$-core inverse. The
$\bra{i}{m}$-core inverse of a complex matrix extends the notions of the core
inverse defined by Baksalary and Trenkler \cite{BT} and the core-EP inverse
defined by Manjunatha Prasad and Mohana \cite{MM}. The $\pare{j}{m}$-core
inverse of a complex matrix extends the notions of the core inverse and the
${\rm DMP}$-inverse defined by Malik and Thome \cite{MT}. Moreover, the
formulae and properties of these two new concepts are investigated by using
matrix decompositions and matrix powers.
