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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)+(2π 2 c 4 -cos2t+2π 2 c 4 +1/4)u(t)=4π 2 c 4 μu(t),

where c is a a given constant while μ 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(2iπν) with Reν[-1/2,1/2). The main results are stated in two theorems which concern the cases ν=-1/2 and ν-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:
34C27Almost and pseudo-almost periodic solutions of ODE
33E10Lamé, Mathieu, and spheroidal wave functions
81Q05Closed and approximate solutions to quantum-mechanical equations