Extensions of Cline's formula for some new generalized inverses

Abstract

Let $a,b,c,d$ be elements in a unital associative ring $\mathcal{R}$. In this note, we generalize Cline's formula for some new generalized inverses such as strong Drazin inverse, generalized strong Drazin inverse, Hirano inverse and generalized Hirano inverse to the case when $acd=dbd$ and $dba=aca$. As a particular case, some recent results are recovered.