Polynomial solutions for a class of second-order linear differential equations

摘要

We analyze the polynomial solutions of the linear differential equation p2(x)y″+p1(x)y′+p0(x)y=0 where pj(x) is a jth-degree polynomial. We discuss all the possible polynomial solutions and their dependence on the parameters of the polynomials pj(x). Special cases are related to known differential equations of mathematical physics. Classes of new soluble problems are exhibited. General results are obtained for weight functions and orthogonality relations.