Newton representation of functions over natural integers having integral difference ratios

@article{Cgielski2013NewtonRO,
  title={Newton representation of functions over natural integers having integral difference ratios},
  author={Patrick C{\'e}gielski and Serge Grigorieff and Ir{\`e}ne Guessarian},
  journal={ArXiv},
  year={2013},
  volume={abs/1310.1507}
}
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) ≡ 0 (mod (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 ↦ ⌊e1/a ax x!⌋, with a ∈ ℤ\{0, 1}, and a function equal to ⌊e x!⌋ except… 
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