# 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).

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.

• 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). Feb 10 at 18:54
• 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. Feb 10 at 20:54
• 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! Feb 10 at 21:11
• the same expansion should hold for all classes of Mathieu function, I added the simple argument (which I should have realized immediately) Feb 11 at 15:33
• 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. Feb 18 at 11:53