  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.
