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

Abstract

We analyze the polynomial solutions of the linear differential equation $p_2(x)y^{\prime \prime }+p_1(x)y^{\prime }+p_0(x)y=0$ where $p_j(x)$ is a $j^{\mathrm{th}}$-degree polynomial. We discuss all the possible polynomial solutions and their dependence on the parameters of the polynomials $p_j(x)$. 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.

23 pages

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