Newton representation of functions over natural integers having integral difference ratios

Abstract : Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying $f(a)-f(b)\equiv0 \pmod { (a-b)}$ for all $a>b$. We characterize this class of functions via their representations as Newton series. This class, which obviously contains all polynomials with integral coefficients, also contains unexpected functions, for instance all functions $x\mapsto\lfloor e^{1/a}\;a^x\;x!\rfloor$, with $a\in\Z\setminus\{0,1\}$, and a function equal to $\lfloor e\;x!\rfloor$ except on $0$. Finally, to study the complement class, we look at functions $\N\to\RR$ which are not uniformly close to any function having integral difference ratios.
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Submitted on : Saturday, October 5, 2013 - 8:44:57 PM
Last modification on : Friday, January 4, 2019 - 5:32:57 PM

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Patrick Cegielski, Serge Grigorieff. Newton representation of functions over natural integers having integral difference ratios. 2013. ⟨hal-00870168⟩

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