Construction of polynomial solutions of a linear second-order differential equation

For given particular polynomial solutions, we construct polynomial coefficients of a second-order differential equation. We obtain a criterion for the equation in question to have two solutions of the form (u_i (t) = a_i t^{varrho _i } + t^{varrho _i - 1} ). Under certain restrictions on the parameters, polynomial solutions of the Heun equation are given.