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)](https://dlmf.nist.gov/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 $a\_\alpha, b\_\beta, c\_\gamma, d\_\delta \neq 0$.




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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)](https://dlmf.nist.gov/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).




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[1] [NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/)



Taylor expansion of Modified Mathieu functions
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](https://homepages.mty.itesm.mx/jgutierr/Mathieu/Mathieu.pdf) 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.




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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.


