Research PapersNo Access
Newton representation of functions over natural integers having integral difference ratios
https://doi.org/10.1142/S179304211550092XCited by:4 (Source: Crossref)
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 ↦ ⌊e1/a ax 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.
AMSC: 11A05, 11B65
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