\(p\)-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 \(e(n)/n\).


12H25 \(p\)-adic differential equations
05A15 Exact enumeration problems, generating functions
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