Consider a complex unital Banach algebra \(\mathcal {A}.\) For \(x_1,x_2,x_3\in \mathcal {A},\) in this paper, we establish that under certain assumptions on \(x_1,x_2,x_3\) , Drazin (resp. g-Drazin) invertibility of any three elements among \(x_1,x_2,x_3\) and \(x_1+x_2+x_3~(\text {or }x_1x_2+x_1x_3+x_2x_3)\) ensure the Drazin (resp. g-Drazin) invertibility of the remaining one. As a consequence for two idempotents \(p,q\in \mathcal {A},\) this result indicates the equivalence between Drazin (resp. g-Drazin) invertibility of \(\lambda _1p+\gamma _1q-\lambda _1pq+\lambda _2\left( pqp-(pq)^2\right) +\cdots +\lambda _m\left( (pq)^{m-1}p-(pq)^m\right) \) and \(\lambda _1-\lambda _1pq+\lambda _2\left( pqp-(pq)^2\right) +\cdots +\lambda _m\left( (pq)^{m-1}p-(pq)^m\right) ,\) where \(\gamma _1,\lambda _i\in \mathbb {C}\) for \(i=1,2,\cdots ,m,\) with \(\lambda _1\gamma _1\ne 0;\) which extend the work of Barraa and Benabdi [1]. Furthermore, for \(x_1,x_2\) , we establish that the Drazin (resp. g-Drazin) invertibility of any two elements among \(x_1,x_2\) and \(x_1+x_2\) indicates the Drazin (resp. g-Drazin) invertibility of the remaining one, provided that \(x_1x_2=\alpha (x_1+x_2)\) for some \(\alpha \in \mathbb {C}\) . Additionally, if it exists, we furnish a new formula to represent the Drazin (resp. g-Drazin) inverse of any element among \(x_1,x_2\) and \(x_1+x_2\) , by using the other two elements and their Drazin (resp. g-Drazin) inverse.
