As stated in the title, my goal is to express a complex exponential functions (i.e. a plane-wave) as the sum of mathieu functions (elliptic functions). In all the rest, I will follow the notation of Abramowitz in "Handbook of Mathematical functions" and Zhang in "Computation of special functions".
It is well known that we can express an exponential as the sum of Bessel functions as follow: \begin{align} e^{-ikx} = e^{-ikr\cos(\theta)} = \sum_{n=-\infty}^{+\infty} (-i)^n J_n(kr) e^{in\theta}. \end{align} In the general case where the plane-wave is coming from an angle $\phi_0$, we have: \begin{align} e^{-ik[x\cos(\phi_0) + y\sin(\phi_0)]} &= e^{-ikr[\cos(\theta)\cos(\phi_0) + \sin(\theta)\sin(\phi_0)]}\\ &= e^{-ikr\cos(\theta-\phi_0)} \\ &= \sum_{n=-\infty}^{+\infty} (-i)^n J_n(kr) e^{in(\theta-\phi_0)}. \end{align}
In an article from Chaos-Cador, L., & Ley-Koo, E. (2002), entitled "Mathieu functions revisited: matrix evaluation and generating functions", it was demonstrated in Eq. (38) that: \begin{align} e^{-ikx} &= e^{-ikc\cosh(u)\cos(v)} = e^{-2i\sqrt{q}\cosh(u)\cos(v)} \\&= 2\sum_{n=0}^{+\infty}(-i)^n ce_n(0, q)ce_n(v, q)Mc_n^{(1)}(u, q) \end{align} and in Eq. (41) that: \begin{align} e^{-iky} &= e^{-ikc\cosh(u)\cos(v)} = e^{-2i\sqrt{q}\sinh(u)\sin(v)} \\& = 2\sum_{r=0}^{+\infty}(-1)^r ce_{2r}\left(\frac{\pi}{2}, q\right)ce_{2r}(v, q)Mc_{2r}^{(1)}(u, q) \\& + 2i\sum_{r=0}^{+\infty}(-1)^r se_{2r+1}\left(\frac{\pi}{2}, q\right)se_{2r+1}(v, q)Ms_{2r+1}^{(1)}(u, q), \end{align} where we have that $2\sqrt{q}=kc$
My problem now is to find an expression for $e^{-ik[x\cos(\phi_0) + y\sin(\phi_0)]}$. I have tried the brutal way by defining $2\sqrt{q_c}=kc\cos(\phi_0)$ and $2\sqrt{q_s}=kc\sin(\phi_0)$ and then multiply both equations together. It works perfectly well even for complex $q$ which is very nice. BUT it is not a pratical equation for what I want to do.
My guess would be that, as in the Bessel case, we need to find a transformation for $x\cos(\phi_0) + y\sin(\phi_0)$ such that we can use the previous equations. Also, using intuition, I tried the following equation: \begin{equation} 2\sum_{n=0}^{+\infty}(-i)^n ce_n(\phi_0, q)ce_n(v, q)Mc_n^{(1)}(u, q) + 2\sum_{n=0}^{+\infty}(-i)^n se_n(\phi_0, q)se_n(v, q)Ms_n^{(1)}(u, q), \end{equation} which would be the perfect equation for me. Interestingly, it works for real values of $q$. However, as soons as $q$ is complex (which I need) it breaks and does not work anymore.
Any idea on how to obtain a pratical equation for complex value of q ?