On the characteristic exponents of Floquet solutions to the Mathieu equation

We study the Floquet solutions of the Mathieu equation. In order to find an explicit relation between the characteristic exponents and their corresponding eigenvalues of the Mathieu operator, we consider the Whittaker-Hill formula. This gives an explicit relation between the eigenvalue and its characteristic exponent. The equation is explicit up to a determinant of an infinite dimensional matrix. We find a third-order linear recursion for which this determinant is exactly the limit. An explicit solution for third-order linear recursions is obtained which enables us to write the determinant explicitly.