Construction of Polynomial Eigenfunctions of a Second-Order Linear Differential Equation

A system of third-order recurrence relations for the coefficients of polynomial eigenfunctions (PEFs) of a differential equation is solved. A recurrence relation for three consecutive PEFs and a formula for differentiating PEFs are obtained. We consider differential equations one of which generalizes the Hermite and Laguerre differential equations and the other is a generalization of the Jacobi differential equation. For these equations, we construct functions bringing them to self-adjoint form and find conditions under which these functions become weight functions. Examples are given where the PEFs for nonweight functions do not have real zeros.