Multiplicative and additive recurrent sequences
An arithmetic function \(g: \mathbb{N}\to \mathbb{C}\) is called recurrent (of order \(k\), \(k\in \mathbb{N}\)), if there exist complex numbers \(a_ 0,\dots, a_{k- 1}\), \(a_ 0\neq 0\), such that \[ g(n+k)+ a_{k-1} g(n+k-1)+\cdots+ a_ 0 g(n)=0 \tag \(*\) \] for all \(n\in \mathbb{N}\). If \((*)\) holds for all sufficiently large natural numbers \(n\) only, then \(g\) is called ultimately recurrent. \textit{A. Sárközy} [Stud. Sci. Math. Hung. 13, 79-104 (1978; Zbl 0481.10001)] resp. \textit{L. Lovász}, \textit{A. Sárközy} and \textit{M. Simonovits} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 24, 205-215 (1981; Zbl 0478.10003)] determined all multiplicative and additive functions which are recurrent. \textit{E. Heppner} and the reviewer [Analysis 5, 87-95 (1985; Zbl 0563.10005) and Analysis 10, 17-21 (1990; Zbl 0707.11069)] modified the proofs in order to find the ultimately recurrent multiplicative and additive functions. They asked for simple direct proofs of the result that for additive and multiplicative functions the classes of recurrent resp. ultimately recurrent functions coincide. Using the shift operator \(z\) defined by \(g\mapsto zg\), \(zg(n)= g(n+1)\) and identifying \((*)\) with its companion polynomial \(f(z)= z^ k+ a_{k-1} z^{k-1}+ \cdots+ a_ 0\) the author gives such proofs. Moreover the same method leads to a shorter way to determine all multiplicative and additive recurrent functions.
scientific article; zbMATH DE number 687026