We use a set of eigendistributions in constructing the formal solution of tho set of integrodifferential equations for the correlation functionals in the context of the Bogoliubov formalism. The equations apply to fully ionized systems interacting through the Coulomb potential, and the method is displayed to an arbitrary order in the interaction strength for the homogeneous case. It is indicated that the method is equally applicable to the slightly inhomogeneous systems for which a perturbation series solution can be developed formally. It is also possible to apply this method in solving the initial-value problem of the BBGKY (Bogoliubov, Born, Green, Kirkwood, Yvon) hierarchy. The eigendistributions employed are those developed by Van Kampen and Case in connection with the linearized Vlasov equation. Although the resulting Green's function is quite complicated, it exhibits the general structure of the solutions.