Multiplicative functions <i>k</i> -additive on hexagonal numbers

Abstract Let k ≥ 3 be an integer. We prove that the set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">H</m:mi> </m:mrow> </m:math> $ \mathcal{H} $ of all nonzero hexagonal numbers is a k -additive uniqueness set for the set of multiplicative functions. That is, if a multiplicative function f k satisfies a multivariate Cauchy functional equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:msub> <m:mi>f</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>+</m:mo> <m:mo>⋯</m:mo> <m:mo>+</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:msub> <m:mi>f</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> <m:mo>+</m:mo> <m:msub> <m:mi>f</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> <m:mo>+</m:mo> <m:mo>⋯</m:mo> <m:mo>+</m:mo> <m:msub> <m:mi>f</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:math> $$ \begin{equation}f_k(x_1+x_2+\cdots+x_k)=f_k(x_1)+f_k(x_2)+\cdots+f_k(x_k) \end{equation} $$ for arbitrary <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>x</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>∈</m:mo> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">H</m:mi> </m:mrow> </m:math> $ x_1,\dots,x_k\in \mathcal{H} $ , then f k is the identity function f k ( n ) = n for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi mathvariant="double-struck">N</m:mi> </m:mrow> </m:math> $ n\in\mathbb{N} $ . This extends the work of Kim et al. for k = 2.