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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

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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.

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  • $\begingroup$ That nice, however, they do power expansion in the parameter $q$ ($c$ in their notations) and I am interested in the power expansion in the variable $\xi$ ($u$ in their notations). $\endgroup$ Feb 10 at 18:54
  • $\begingroup$ indeed, I misread the notation; I have now calculated the series expansion explicitly for one particular case; as you see, the lack of a closed form expression for the Fourier coefficients limits the usefulness of the expansion. $\endgroup$ Feb 10 at 20:54
  • $\begingroup$ I was particularly interested in the value of the order, so we get $\alpha_{2n}^{(1)} = 0$ and $\beta_{2n}^{(1)} = 2$ if $a_{2n}(q) \neq q$. Thank you! $\endgroup$ Feb 10 at 21:11
  • $\begingroup$ the same expansion should hold for all classes of Mathieu function, I added the simple argument (which I should have realized immediately) $\endgroup$ Feb 11 at 15:33
  • $\begingroup$ Your second method is fast to compute the Taylor expansion, however it is not so useful for me since they do not tell me if $w(0) = 0$ or not. $\endgroup$ Feb 18 at 11:53

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