Algebraic methods to compute Mathieu functions

The standard form of the Mathieu differential equation is y"(z) + (a - 2q c os 2z) y(z) = 0, where a is the characteristic number and q is a real param eter. The most useful solution forms are given in terms of expansions for e ither small or large values of q. In this paper we obtain closed formulae f or the generic term of expansions of Mathieu functions in the following cas es: (1) standard series expansion for small q; (2) Fourier series expansion for small q; (3) asymptotic expansion in terms of trigonometric functions for large q an d (4) asymptotic expansion in terms of parabolic cylinder functions for large q. We also obtain closed formulae for the generic term of expansions of charac teristic numbers and normalization formulae for small and large q. Using th ese formulae one can efficiently generate high-order expansions that can be used for implementation of the algebraic aspects of Mathieu functions in c omputer algebra systems. These formulae also provide alternative methods fo r numerical evaluation of Mathieu functions.