On multiplicative and additive differences

Summary.

This paper is about the characterization of those functions which can be expressed as a sum of a generalized polynomial and a generalized logarithmic polynomial. The simplest case of generalized polynomials and generalized logarithmic polynomials of the first degree is to characterize those functions which are sums of affine and logarithmic functions. This problem is solved in [4]. The main result of this article is the following characterization theorem: A function $f\,:\,\mathbb{R}_ + \to \mathbb{R}$ is the sum of a generalized polynomial of degree at most n and of a generalized logarithmic polynomial of degree at most m if and only if all of its m-th multiplicative differences are generalized polynomials of degree at most n.