Taylor expansion of Modified Mathieu functions Do we know the Taylor expansion at $0$ of the radial Mathieu functions $(\mathsf{Mc}_n^{(j)}(\,\cdot\,, \sqrt{q}))_{n \ge 0}$ and $(\mathsf{Ms}_n^{(j)}(\,\cdot\,, \sqrt{q}))_{n \ge 1}$, for $q \in \mathbb{R}$ and $j \in \{1, 2, 3, 4\}$, with the definition and conventions of [Sec. 28.20(iv), 1]? I could not find it in [1] or their references and I am not able to find a way to deduce it from other expansions.
To be more precise, I am interested in the values of the first two orders $\alpha < \beta$ and $\gamma < \delta$, with respect to $n$, $q$, and $j$, in the Taylor expansions
$$
\mathsf{Mc}_n^{(j)}(\xi, \sqrt{q}) = a_\alpha \xi^\alpha + b_\beta \xi^\beta + o(\xi^\beta)
\quad \text{and} \quad
\mathsf{Ms}_n^{(j)}(\xi, \sqrt{q}) = c_\gamma \xi^\gamma + d_\delta \xi^\delta + o(\xi^\delta)
$$
as $\xi \to 0^+$ with $a_\alpha, b_\beta, c_\gamma, d_\delta \neq 0$.

To recall, The radial Mathieu functions $(\mathsf{Mc}_n^{(j)}(\,\cdot\,, \sqrt{q}))_{n \ge 0}$ are solutions of the ordinary differential equation
$$
w''(\xi) - (a_n(q) - 2q \cosh(2\xi))w(\xi) = 0 \tag{1}
$$
where $a_n(q)$ are the even eigenvalues of $-g'' + 2q\cos(2\eta)\, g$ on $\mathbb{R} / 2\pi\mathbb{Z}$ [28.2(v), 1], such that
$$
\mathsf{Mc}_n^{(j)}(\xi, \sqrt{q})
= \mathcal{C}_n^{(j)}(2\sqrt{q} \cosh(\xi)) + O(\cosh(\xi)^{-1}),
\quad \text{as } \xi \to +\infty \tag{2}
$$
where $\mathcal{C}_n^{(1)} = \mathsf{J}_n$, $\mathcal{C}_n^{(2)} = \mathsf{Y}_n$, $\mathcal{C}_n^{(3)} = \mathsf{H}_n^{(1)}$, and $\mathcal{C}_n^{(4)} = \mathsf{H}_n^{(2)}$, the Bessel and Hankel functions. The radial Mathieu functions $(\mathsf{Ms}_n^{(j)}(\,\cdot\,, \sqrt{q}))_{n \ge 1}$ satisfy the same ODE (1) but with $b_n(q)$ the odd eigenvalues instead of $a_n(q)$ the even eigenvalues and satisfy the same relations (2).

[1] NIST Digital Library of Mathematical Functions
 A: I have not found the series expansion worked out explicitly, but it can be obtained from the representation of the Mathieu functions as series of Bessel functions. I found this collection of formulas convenient.
There are four classes of radial Mathieu functions, corresponding to series of the four types of Bessel functions ($J,N,I,K$ labeled by $j=1,2,3,4$). Each class has two varieties $Mc_{n}^{(j)}$ and $Ms_{n}^{(j)}$. For definiteness, let me consider $Mc_{2n}^{(1)}$, which has an expansion in terms of Bessel functions $J_{2k}$, see eq. 2.25a in the cited source:
$$Mc_{2n}^{(1)}(\xi,\sqrt{q})=\frac{\sum_{s=0}^\infty A_{2s}(q;2n)}{A_0(q;2n)}\sum_{k=0}^\infty A_{2k}(q;2n)J_{2k}(2\sqrt{q}\sinh \xi).$$
The first two terms in the power series in $\xi$ are
$$Mc_{2n}^{(1)}(\xi,\sqrt{q})=\frac{\sum_{s=0}^\infty A_{2s}(q;2n)}{A_0(q;2n)}\bigl[A_0(q;2n)+q\xi^2 \left(\tfrac{1}{2}A_2(q;2n)-A_0(q;2n) \right)+{\cal O}(\xi^4)\bigr].$$
The Fourier coefficients $A_{2s}(q;2n)$ need to be determined by a recursion relation, they have no closed form expression. The relation $a_{2n}(q)A_0(q;2n)=qA_2(q;2n)$ allows to simplify the series expansion as
$$Mc_{2n}^{(1)}(\xi,\sqrt{q})=\left(\sum_{s=0}^\infty A_{2s}(q;2n)\right)\bigl[1+\xi^2 \left(\tfrac{1}{2}a_{2n}(q)-q \right)+{\cal O}(\xi^4)\bigr].$$
This at least gives you explicitly the ratio of the zeroth and second order term.

Note that this could have been obtained directly from the differential equation, by expanding
$$w''(\xi)=[a-2q+{\cal O}(\xi^2)]w(\xi)\Rightarrow w(\xi)=w(0)[1+(\tfrac{1}{2}a-q)\xi^2+{\cal O}(\xi^4)].$$
This implies that the same expansion applies as well to the other classes of radial Mathieu functions.
