The generalized and pseudo $n$-strong Drazin inverse of the sum of elements in Banach algebras

In this paper, we begin by introducing some necessary and sufficient conditions for generalized $n$-strong Drazin invertibility (g$n$s-invertibility) and pseudo $n$-strong Drazin invertibility (p$n$s-invertibility) of an element in a Banach algebra for $n\in \mathbb{N}$. Subsequently, these results are utilized to prove some additive properties of g$n$s (p$n$s)-Drazin inverse in a Banach algebra. This process produces a generalization of some recent results of H Chen, M Sheibani (Linear and Multilinear Algebra $\mathbf{\text{70.1}}$ (2022): 53-65) for g$n$s and p$n$s-Drazin inverse. Furthermore, we define and characterize weighted g$n$s and weighted p$n$s-Drazin inverse in a Banach algebra.