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In this paper, we give new characterizations for the eigenvalues of the prolate wave equation as limits of the zeros of some families of polynomials: the coefficients of the formal power series appearing in the solutions near 0, 1, or ∞ (in the variables x,x−1, or 1/x, respectively). The result, which seems to be true for all values of the parameter τ, according to our numerical experiments, is here proved for small values of the parameter τ.

International audienceWe describe a general procedure for computing Stokes matrices for solutions of linear differential equations with polynomial coefficients. The algorithms developed make an automation of the calculations possible, for a wide class of equations. We apply our techniques to some classical holonomic functions and also for some new special functions that are interesting in their own right: Ecalle's accelerating functions