Integer-valued Euler–Jackson’s finite differences

Abstract

Let D be a Dedekind domain and q be a nonzero element of D which is not a root of unity. We are interested here in the (D)-algebra \({{{\rm Int}^{[k]}_J}(S,D)}\) formed by the polynomials whose values on the subset \({S=\{q^n\mid n\in \mathbb{N}\}}\) belong to D as well as the values of their k first Euler–Jackson’s finite differences. Mainly, we describe the characteristic ideals of \({{{\rm Int}^{[k]}_J}(S,D)}\) and, in the case where D = V is a discrete valuation domain, we give bases of the V-module \({{{\rm Int}^{[k]}_J}(S,V)}\) . Moreover, the description of the maximal ideals of the D-algebra \({{{\rm Int}^{[k]}_J}(S,D)}\) allows us to characterize the subsets S such that \({{\rm Int}_J^{[\infty]}(S,D)}\) is equal to the D-algebra \({{\rm Int}^{(\infty)}(S,D)}\) formed by the polynomials whose values on S belong to D as well as the values of all their derivatives.