Lanczos iterations as well as power series ansatzs often transform our zero-order Hamiltonians into Hessenberg (= tridiagonal plus triangular) matrices.
Then, the single zero order wavefunction becomes only defined in terms of Hill determinants.
Two possible related appropriate recurrent modifications of the textbook Rayleigh-Schrödinger perturbation recipe are outlined.
Published in Int. J. Mod. Phys. A 12 (1997) 299 - 304.