Drazin and g-Drazin invertibility of combinations of three Banach algebra elements

Consider a complex unital Banach algebra A. For x₁,x₂,x₃, in this paper, we establish that under certain assumptions on x₁,x₂,x₃, Drazin (resp. g-Drazin) invertibility of any three elements among x₁,x₂,x₃ and x₁+x₂+x₃ (or x₁x₂+x₁x₃+x₂x₃) ensure the Drazin (resp. g-Drazin) invertibility of the remaining one. As a consequence for two idempotents p,q, this result indicates the equivalence between Drazin (resp. g-Drazin) invertibility of λ₁p+γ₁q-λ₁pq+λ₂(pqp-(pq)²)+⋯+λₘ((pq)ᵐ⁻¹p-(pq)ᵐ) and λ₁-λ₁pq+λ₂(pqp-(pq)²)+⋯+λₘ((pq)ᵐ⁻¹p-(pq)ᵐ), where γ₁,λᵢ for i=1,2,⋯,m, with λ₁γ₁≠0. Furthermore, for x₁,x₂, we establish that the Drazin (resp. g-Drazin) invertibility of any two elements among x₁,x₂ and x₁+x₂ indicates the Drazin (resp. g-Drazin) invertibility of the remaining one, provided that x₁x₂=α(x₁+x₂) for some α. Additionally, if it exists, we furnish a new formula to represent the Drazin (resp. g-Drazin) inverse of any element among x₁,x₂ and x₁+x₂, by using the other two elements and their Drazin (resp. g-Drazin) inverse.