The Wikipedia article (http://en.wikipedia.org/wiki/Mathieu_function) is quite extensive, I believe, and in general there is NO way to explicitly calculate either eigenvalues or eigenfunctions of the Mathieu equation. 

The Wikipedia article also explicitly states that the solutions generally cannot be expressed in terms of hypergeometric functions. 

You can do perturbation theory around $q=\infty$ and recursively calculate asymptotic expansions of eigenfunctions and eigenvalues to any order (semiclassical analysis, basically). However, you seem not to be interested in this. (A reference would be a quite old Paper, unfortunately in german: J. Meixner, "Asymptotische Entwicklung der Eigenwerte und Eigenfunktionen der Differentialgleichungen der Sphäroid-Funktionen und der Mathieuschen Funktionen", Zeitschrift füt angewandte Mathematik und Mechanik, 1948)