This paper presents some additive results on the Drazin inverse and generalized Drazin inverse in a complex Banach algebra \(\mathcal {A}\) . More precisely, it is shown that if \(a, b\in \mathcal {A}\) satisfy that \(ab(a+b) = (a+b)ab\) and ab is quasi-nilpotent (resp., nilpotent), then any two of the generalized Drazin invertibility (resp., Drazin invertibility) of a, b and \(a+b\) imply the remaining one. In this case, by using the uniqueness of the Laurent series of their resolvents expanding in a neighborhood of 0, we obtain an explicit formula that computes the (generalized) Drazin inverse of a (resp., b) in terms of that of \(a+b\) . We also express the (generalized) Drazin inverse of \(a + b\) via those of a and b under some slightly stronger conditions. As applications, we find some new representations for the Drazin inverse of a block complex matrix with certain prescribed conditions.
