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Journal Articles International Journal of Number Theory Year : 2015

Newton representation of functions over natural integers having integral difference ratios

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Patrick Cégielski
  • Function : Author
Serge Grigorieff
  • Function : Author

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) ≡ 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 ↦ ⌊e 1/a a x x!⌋, with a ∈ ℤ\{0, 1}, and a function equal to ⌊e x!⌋ except on 0. Finally, to study the complement class, we look at functions ℕ → ℝ which are not uniformly close to any function having integral difference ratios.

Dates and versions

hal-03774324 , version 1 (10-09-2022)

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Patrick Cégielski, Serge Grigorieff, Irène Guessarian. Newton representation of functions over natural integers having integral difference ratios. International Journal of Number Theory, 2015, 11 (07), pp.2109-2139. ⟨10.1142/S179304211550092X⟩. ⟨hal-03774324⟩
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