Recursive solutions of Lagrangian perturbation theory

In the standard perturbation theory (SPT) of self-gravitating Newtonian fluid in an expanding universe, recurrence relations for higher-order solutions are well known and play an important role both in practical applications and in theoretical investigations. The recurrence relations in Lagrangian perturbation theory (LPT), however, have not been known for a long time. Recently, two different kinds of recurrence relations in LPT have been proposed in limited cases. In this paper, we generalize those methods, and most generally derive the recurrence relations, which are capable of including any initial condition in general models of cosmology. The fastest-growing modes in the general relations are identified, and simplified recurrence relations with accurate approximation for the time dependence are obtained.