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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 $$u''(t)+ (2\pi^2 c^4- \cos 2t+ 2\pi^2 c^4+ 1/4)u(t)= 4\pi^2 c^4\mu u(t),$$ 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 $\exp(2i\pi\nu)$ with $\text{Re }\nu\in [-1/2,1/2)$. 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.

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