Derivation of the generalized Langevin equation by a method of recurrence relations

The generalized Langevin equation was first derived by Mori using the Gram-Schmidt orthogonalization process. This equation can also be derived by a method of recurrence relations. For a physical space commonly used in statistical mechanics, the recurrence relations are simple and they lead directly to the Langevin equation. The Langevin equation is shown to be composed of one homogeneous and one inhomogeneous equation.