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Resummation of formal seriesSolutions of second order linear complex ordinary differential equations near singularities Summary. Formulae for solutions of complex ordinary differential equations in the neighbourhood of irregular singularities contain almost every time divergent series. The Resummation Theory developed in the field of Analytic Functional Equations by J.P. Ramis provides us with a tool-box to perform in different ways some effective calculations and to compare their results. We take full advantage of the possibilities of Computer Algebra, especially of exact rational evaluation.Les travaux r6eents de B. Malgrange, J.P. Ramis, J. Della Dora, C. Dicrescenzo, D. Duval, et E. Tournier ont permis de "calculer" ees solutions par des algorithmes formels. I1 6tait indispensable de calculer ces solutions "exactement", cause de leur comportement 6ventuellement explosif. D'autre part, les 6tudes r6centes de J.P. Ramis, prolongeant les travaux classiques de Borel, Nevanlinna, Watson .... ont permis de passer du traitement de type "sommation au plus petit terme des s6ries asymptotiques" dans de "petits secteurs" du plan complexe /t des resommations dans des "grands secteurs" des s6ries a(x) en g6n6ral divergentes. Cette resommation est vraiment effective quand ai(x ) est une s6rie ksommable, si keQ. Ceci est toujours le cas pour les 6quations du second ordre, notamment pour les 6quations des fonctions sp6ciales. Par contre, dans le cas

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.