Integral difference ratio functions on integers

Abstract

Various problems lead to the same class of functions from integers to integers: functions having integral difference ratio, i.e. verifying f(a) − f(b) ≡ 0 (mod (a − b)) for all a > b. In this paper we characterize this class of functions from Z to Z via their `a la Newton series expansions on a suitably chosen basis of polynomials (with rational coefficients). We also exhibit an example of such a function which is not polynomial but Bessel like.