Investigations in the theory of \(q\)-additive and \(q\)-multiplicative functions. II
type: Item
instance of: scholarly article
title: Investigations in the theory of \(q\)-additive and \(q\)-multiplicative functions. II
zbMATH Open document ID: 1012.11008
DOI: 10.1023/A:1020858928794
published in: Acta Mathematica Hungarica
publication date: 2003-05-18T00:00:00Z
review text: The authors continue their investigations started in [ibid. 91, 53-78 (2001; Zbl 0980.11001)]. To formulate their new results obtained in the present paper, some definitions are needed. Let \(q\geq 2\) be a fixed integer, \({\mathbb A}=\{0,1,\hdots,q-1\}\), and denote by \({\mathbb N}_0\) the set of nonnegative integers. Each \(n\in {\mathbb N}_0\) has a unique \(q\)-ary expansion \[ n=\sum\limits_{j=0}^\infty \varepsilon_j(n)q^j,\;\;\;\varepsilon_j(n)\in{\mathbb A}. \] Let \(G\) be an arbitrary Abelian group. A function \(\varphi:{\mathbb N}_0\to G\) is called a \(G\) valued \(q\)-additive function if \(\varphi(0)=0\) and for every \(n\in {\mathbb N}_0\) \[ \varphi(n)=\sum\limits_{j=0}^\infty \varphi(\varepsilon_j(n)q^j) \] holds. Denote by \({\mathcal A}_q(G)\) the set of such functions. If \(G={\mathbb R}\), then simply write \({\mathcal A}_q={\mathcal A}_q({\mathbb R})\). Moreover, the function \(g:{\mathbb N}_0\to {\mathbb C}\) is called \(q\)-multiplicative, if \(g(0)=1\) and for every \(n\in {\mathbb N}_0\) we have \[ g(n)=\prod\limits_{j=0}^\infty g(\varepsilon_j(n)q^j). \] The set of \(q\)-multiplicative functions is denoted by \({\mathcal M}_q\), and the set of such functions with \(|g(n)|=1\) \((n\in {\mathbb N}_0)\) by \(\bar{{\mathcal M}_q}\). Let \((1\leq)a_1<a_2<\hdots<a_k(<q)\) be mutually coprime integers, each of which is coprime to \(q\). Let \(f_1,\hdots,f_k\in{\mathcal A}_q\), \(g_1,\hdots,g_k\in\bar{{\mathcal M}_q}\) and put \[ l(n)=f_1(a_1n)+\dots+f_k(a_kn),\;\;\;t(n)=g_1(a_1n)\dots g_k(a_kn). \] The authors give necessary and sufficient conditions for the existence of the limit distribution of \(l(n)\), and of the mean value of \(t(n)\).
reviewed by: Lajos Hajdu
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11A67
Mathematics Subject Classification ID: 11K65
zbMATH DE Number: 1912073
zbMATH Keywords: \(q\)-additive and \(q\)-multiplicative functions
zbMATH Keywords: limit distribution
author: K.-H. Indlekofer
author: Imre Kátai
MaRDI profile type: Publication
full work available at URL: a:1020858928794
OpenAlex ID: W4231470388
Recommended article: Investigations in the theory of \(q\)-additive and \(q\)-multiplicative functions. I
Recommended article: On \(q\)-additive functions. II
Recommended article: Q3117612
Recommended article: Q4407940
Recommended article: Q3749115
Recommended article: Some relations among multiplicative and \(q\)-additive functions
Recommended article: On \(q\)-quasiadditive and \(q\)-quasimultiplicative functions
Recommended article: Q3695322
Recommended article: On \(p\)- and \(q\)-additive functions
Recommended article: Q4415413
description: scientific article; zbMATH DE number 1912073