Generalization of Resonant Equations for the Laguerre- and Legendre-Type Polynomials to Equations of the Fourth Order

Abstract

A recurrence algorithm for finding particular solutions of the resonant equation of the fourth order connected with the generalization of Laguerre and Legendre polynomials is constructed and substantiated. For this purpose, we use the general theorem on the representation of particular solutions of resonant equations in Banach spaces proved by Makarov in 1976. An example of the general solution of resonant equations with differential operator for the Laguerre-type polynomials is presented.