Let S be a semigroup and K a field. Ebanks showed recently how the functional equation <Equation ID="Equ7"> <EquationSource Format="TEX">$\begin{aligned} g(xyz) = g(x)g(yz) + g(y)g(xz) + g(z)g(xy) - 2g(x)g(y)g(z), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mn>2</mn> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$g:S \rightarrow K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> denotes the unknown function and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$x,y,z \in S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation>, is related to the sine addition law on S. We simplify Ebanks’ treatment. Furthermore we discuss for solutions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f,g:S \rightarrow K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> of the sine addition law the uniqueness of the component f given g, and we show that f and g are abelian, when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f \ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.

The sine addition law applied to a cosine equation

期刊论文
出版模式：

开放获取
发表日期：

2026年1月13日

查看全文

Henrik Stetkær

摘要

Let S be a semigroup and K a field. Ebanks showed recently how the functional equation $\begin{aligned} g(xyz) = g(x)g(yz) + g(y)g(xz) + g(z)g(xy) - 2g(x)g(y)g(z), \end{aligned}$ $\begin{matrix} g (x y z) = g (x) g (y z) + g (y) g (x z) + g (z) g (x y) - 2 g (x) g (y) g (z), \end{matrix}$ where $g:S \rightarrow K$ $g : S \to K$ denotes the unknown function and $x,y,z \in S$ $x, y, z \in S$ , is related to the sine addition law on S. We simplify Ebanks’ treatment. Furthermore we discuss for solutions $f,g:S \rightarrow K$ $f, g : S \to K$ of the sine addition law the uniqueness of the component f given g, and we show that f and g are abelian, when $f \ne 0$ $f \neq 0$ .