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On the growth of convergence radii for the eigenvalues of the Mathieu equation. (English) Zbl 0912.34027
The author reconsiders the classical Mathieu equation $$y''+ (\mu+ 2\lambda\cos 2x) y=0$$ in the restricted case where it has nontrivial $\pi$-periodic or $\pi$-antiperiodic solutions. In the first case it holds $y(x+\pi)= y(x)$ while in the second case $y(x+ \pi)= -y(x)$, yielding $\mu$ as a function of $\lambda$, say $\mu= \mu_0(\lambda)$ $(n= 1,2,\dots)$ (eigenvalues). The aim is to find a new, sharper estimation for the radius of convergence of the power series of $\mu_n(\lambda)$ about the point $\lambda= 0$. The result, which seems to be the best one, is as follows: $$\lim_{n\to\infty}\inf {\rho_n\over n^2}\ge k k' K^2= 2.041834\dots\ .$$ Here, $\rho_n$ refers the radius of convergence of power series of $\mu_0(\lambda)$ while $K= K(k)$ denotes the complete elliptic integral of the first kind and $k'=(1- k^2)^{1/2}$. As to the modulus $k$, it is determined through the relation $2E= K$, $E$ being the corresponding complete elliptic integral of the second kind.

##### MSC:
 34B30 Special ODE (Mathieu, Hill, Bessel, etc.) 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators 33E10 Lamé, Mathieu, and spheroidal wave functions
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