## $$p$$-adic analysis and classical sequences of numbers. (Analyse $$p$$-adique et suites classiques de nombres.)(French)Zbl 0977.12500

Summary: Let $$(a(n))$$ $$(n=0,1,...)$$ be a sequence of rational numbers (or more generally of algebraic numbers over $$\mathbb{Q}$$) and let $$p$$ be a prime number. We show that the property for the sequence $$(a(n))$$ to be periodical mod any $$h$$-th power of $$p$$ after a certain $$n$$ is equivalent to some $$p$$-adic analytical continuation properties that the generating function $$a$$ for the sequence $$(a(n))$$ has to hold. 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 to 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 sequence $$(e(n))$$ 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 $$e(n)/n$$.

### MSC:

 12H25 $$p$$-adic differential equations 05A15 Exact enumeration problems, generating functions
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