On the Generalized Ince Equation

Abstract

We investigate a Hill differential equation with trigonometric polynomial coefficients. we are interested in solutions which are even or odd and have period π or semi-period π. With one of the mentioned boundary conditions, our equation constitute a regular Sturm-Liouville eigenvalue problem. Using Fourier series representation each one of the four Sturm-Liouville operators is represented by an infinite banded matrix. In the particular cases of Ince and Lam equations, the four infinite banded matrices become tridiagonal. We then investigate the problem of coexistence of periodic solutions and that of existence of polynomial solutions.