In this paper, we introduce the notion of $q$-quasiadditivity of arithmetic functions, as well as the related concept of $q$-quasimultiplicativity, which generalise strong $q$-additivity and -multiplicativity, respectively. We show that there are many natural examples for these concepts, which are characterised by functional equations of the form $f(q^{k+r}a + b) = f(a) + f(b)$ or $f(q^{k+r}a + b) = f(a) f(b)$ for all $b < q^k$ and a fixed parameter $r$. In addition to some elementary properties of $q$-quasiadditive and $q$-quasimultiplicative functions, we prove characterisations of $q$-quasiadditivity and $q$-quasimultiplicativity for the special class of $q$-regular functions. The final main result provides a general central limit theorem that includes both classical and new examples as corollaries.
@article{10_37236_6373,
author = {Sara Kropf and Stephan Wagner},
title = {On \(q\)-quasiadditive and \(q\)-quasimultiplicative functions},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {1},
doi = {10.37236/6373},
zbl = {1405.11005},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6373/}
}
TY - JOUR
AU - Sara Kropf
AU - Stephan Wagner
TI - On \(q\)-quasiadditive and \(q\)-quasimultiplicative functions
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/6373/
DO - 10.37236/6373
ID - 10_37236_6373
ER -
%0 Journal Article
%A Sara Kropf
%A Stephan Wagner
%T On \(q\)-quasiadditive and \(q\)-quasimultiplicative functions
%J The electronic journal of combinatorics
%D 2017
%V 24
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/6373/
%R 10.37236/6373
%F 10_37236_6373
Sara Kropf; Stephan Wagner. On \(q\)-quasiadditive and \(q\)-quasimultiplicative functions. The electronic journal of combinatorics, Tome 24 (2017) no. 1. doi: 10.37236/6373
