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) ≡ 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.