Polynomial functions on rings of dual numbers over residue class rings of the integers

The ring of dual numbers over a ring R is R[α] = R[x]/(x2), where α denotes x + (x2). For any finite commutative ring R, we characterize null polynomials and permutation polynomials on R[α] in terms of the functions induced by their coordinate polynomials (f1, f2 ∈ R[x], where f = f1 + αf2) and their formal derivatives on R. We derive explicit formulas for the number of polynomial functions and the number of polynomial permutations on Zpn[α] for n ≤ p (p prime).