[Formerly: Publ. I.R.M.A. Strasbourg, 1982, 182/S-04, p. 1-21.]

It is shown how the geometry of the domain on which the ordinary
generating function *a* is a *p*-adic analytic element gives a strong
indication for the congruences bo be satisfied by the *a*(*n*)'s.
Furthermore, if the exponential generating function *A* for the
a(n)'s satisfies certain functional properties, then *a* is a *p*-adic
analytic element on a domain containing the open disk of center 0
and radius 1. This is the case when *A* satisfies an algebraic
differential equation and if *a*(*n*) is integral or if the reciprocal
of *A* possesses certain properties.

We show how we can obtain explicit results on some classical
sequences of numbers. We are led to introduce the formal Laplace
transform that maps *A* onto a and derive a few straightforward
properties. Finally, we show the link between the congruences of
Cartier type satisfied by a sequences (*e*(*n*)) of integers and the
congruences of Kummer type satisfied by the coefficients *a*(*n*) of
an exponential generating function that is the
reciprocal of the ordinary generating function for the *e*(*n*)/*n*.

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