On multiplicative and additive differences (Q1774143)

The authors deal with the characterization of 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 was solved by \textit{Z. Daróczy, K. Lajkó} and \textit{L. Székelyhidi} [Publ. Math. Debrecen 24, 173--179 (1977; Zbl 0372.39004)]. The main result is the following characterization theorem: A function \(f: R \to R\) 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 \(n\).