Let an unperturbed Hamiltonian T possess a tridiagonal matrix form.
Make a trick: Choose any E_0 and "correct" H_0 = T + U with an "ad hoc" separable U.
Wavefunctions become easily defined (by recurrences) in any order (incl. zero),
the current "unperturbed" diagonalization of H_0 not necessary,
replaced by the (much simpler) inversion of T,
using, say, continued fractions.
A modified textbook Rayleigh-Schrödinger perturbation recipe obtained,
Phys. Lett. A 120 (1987) 317 - 321,
Phys. Rev. A 35 (1987) 2448.