Left and right resolvents of left and right generalized Drazin invertible operators are introduced in this paper. The construction of left and right resolvents allows us to find, in terms of the coefficients of Laurent series, new representation results for left and right generalized Drazin inverses and the associated spectral projections. Fundamental characterizations of left and right generalized Drazin invertible operators are also obtained, using essentially the range, the quasi-nilpotent part and the analytic core.