Given a semigroup S equipped with an involutive automorphism <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>, we determine the complex-valued solutions f, g, h of the functional equation <Equation ID="Equ83"> <EquationSource Format="TEX">$\begin{aligned}f(x\sigma (y))=f(x)g(y)+g(x)f(y)+h(x)h(y),\,\,x,y\in S,\end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>S</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in terms of multiplicative functions and solutions of the special cases of sine and cosine–sine functional equations <Equation ID="Equ84"> <EquationSource Format="TEX">$\begin{aligned} \varphi (xy)=\varphi (x)\chi (y)+\chi (x)\varphi (y), x,y\in S \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>S</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>and <Equation ID="Equ85"> <EquationSource Format="TEX">$\begin{aligned} \psi (xy)=\psi (x)\chi (y)+\chi (x)\psi (y)+\varphi (x)\varphi (y), x,y\in S \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>S</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\chi :S\rightarrow \mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>χ</mi> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> is a multiplicative function.

A generalization of the cosine-sine functional equation on a semigroup

期刊论文
发表日期：

2026年3月14日

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Omar Ajebbar,
Elhoucien Elqorachi

摘要

Given a semigroup S equipped with an involutive automorphism $\sigma $ $σ$ , we determine the complex-valued solutions f, g, h of the functional equation $\begin{aligned}f(x\sigma (y))=f(x)g(y)+g(x)f(y)+h(x)h(y),\,\,x,y\in S,\end{aligned}$ $\begin{matrix} f (x σ (y)) = f (x) g (y) + g (x) f (y) + h (x) h (y), x, y \in S, \end{matrix}$ in terms of multiplicative functions and solutions of the special cases of sine and cosine–sine functional equations $\begin{aligned} \varphi (xy)=\varphi (x)\chi (y)+\chi (x)\varphi (y), x,y\in S \end{aligned}$ $\begin{matrix} φ (x y) = φ (x) χ (y) + χ (x) φ (y), x, y \in S \end{matrix}$ and $\begin{aligned} \psi (xy)=\psi (x)\chi (y)+\chi (x)\psi (y)+\varphi (x)\varphi (y), x,y\in S \end{aligned}$ $\begin{matrix} ψ (x y) = ψ (x) χ (y) + χ (x) ψ (y) + φ (x) φ (y), x, y \in S \end{matrix}$ where $\chi :S\rightarrow \mathbb {C}$ $χ : S \to C$ is a multiplicative function.