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Mathieu functions and Klein-Gordon polynomials. (Fonctions de Mathieu et polynômes de Klein-Gordon.) (French) Zbl 0912.34035

The author considers the Mathieu differential equation

${u}^{\text{'}\text{'}}\left(t\right)+\left(2{\pi }^{2}{c}^{4}-cos2t+2{\pi }^{2}{c}^{4}+1/4\right)u\left(t\right)=4{\pi }^{2}{c}^{4}\mu u\left(t\right),$

where $c$ is a a given constant while $\mu$ is the (known) spectral parameter, and gives some explicit expressions for the Fourier coefficients of its quasi-periodic solutions corresponding to the characteristic multiplier $exp\left(2i\pi \nu \right)$ with $\text{Re}\phantom{\rule{4.pt}{0ex}}\nu \in \left[-1/2,1/2\right)$. The main results are stated in two theorems which concern the cases $\nu =-1/2$ and $\nu \ne -1/2$, respectively. The results involve certain sets of polynomials which have connections with the Klein-Gordon equation. Notice that the Fourier coefficients in question satisfy a three-term recurrence relation whose explicit solution is not known in terms of known functions.

##### MSC:
 34C27 Almost and pseudo-almost periodic solutions of ODE 33E10 Lamé, Mathieu, and spheroidal wave functions 81Q05 Closed and approximate solutions to quantum-mechanical equations