On a functional equation related to roots of translations of positive integers

Abstract

© 2014, Springer Basel. We consider the functional equation f q (n) = f(n + 1) + k, where $${q \geqslant 2}$$q⩾2 and $${k \in \mathbb{Z}}$$k∈Z are given, and $${f :\mathbb{N} \to \mathbb{N}}$$f:N→N. This functional equation is related to roots of translations of positive integers, and another motivation for studying this functional equation is the fact that it can be thought of as the “prototypical case” of a more general functional equation of a very broad scope. Our main result is that the considered functional equation has a solution if and only if either k = 0 or $${k \geqslant -1}$$k⩾-1 and $${q - 1\mid k + 1}$$q-1∣k+1. We further find all solutions for the case q = 3 and k = 1, which is an example that illustrates that the considered functional equation can have a very unexpected set of solutions even with quite small parameters.