Mathieu function $\operatorname{ce}_m(q,x)$ and $\operatorname{se}_m(q,x)$ are periodic solution of Mathieu equation, which is analogous to $\cos(x)$ and $\sin(x)$.
However, unlike $\cos(x)$ and $\sin(x)$, the derivative of Mathieu function $\operatorname{ce}^\prime_m(q,x)$ and $\operatorname{se}^\prime_m(q,x)$ are not equal to $-\operatorname{se}_m(q,x)$ and $\operatorname{ce}_m(q,x)$.
My question is, how to express Mathieu function's first derivative in terms of Mathieu function? Explicitly, in the following expansion, what is the analytical expression of $A_m^k$(not in the integral form)? $$ \operatorname{ce}^\prime_m(q,x) = \sum_{k}a_m^n \operatorname{se}_n(q,x) $$
If no analytical expression is available, does anyone known the existing numerical library which can generate these coefficients conveniently?