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

TL;DR

This paper investigates polynomial solutions of second-order linear differential equations with polynomial coefficients, exploring their parameter dependence, special cases, and orthogonality properties, and introduces new solvable problems in mathematical physics.

Contribution

It provides a comprehensive analysis of polynomial solutions for a broad class of differential equations, including new solvable cases and general orthogonality results.

Findings

Classified all polynomial solutions based on parameters.

Identified new solvable differential equations related to physics.

Derived general orthogonality relations for solutions.

Abstract

We analyze the polynomial solutions of the linear differential equation $p_2(x)y''+p_1(x)y'+p_0(x)y=0$ where $p_j(x)$ is a $j^{\rm th}$-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.