Let S be a semigroup, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sigma :S \rightarrow S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation> a surjective homomorphism and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">F</mi> </math></EquationSource> </InlineEquation> a field of characteristic different from 2. We study solutions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f,g :S \rightarrow \mathbb {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">F</mi> </mrow> </math></EquationSource> </InlineEquation> of the functional equation <Equation ID="Equ23"> <EquationSource Format="TEX">$\begin{aligned} f(x\sigma (y)) = f(x)g(y) + \beta g(x)f(y) + \gamma f(x)f(y), \ x,y \in S, \end{aligned}$</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>β</mi> <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>γ</mi> <mi>f</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> <mspace width="4pt" /> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>S</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in which <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\beta \in \mathbb {F}^*$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">F</mi> </mrow> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\gamma \in \mathbb {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>∈</mo> <mi mathvariant="double-struck">F</mi> </mrow> </math></EquationSource> </InlineEquation> are constants. The functional equation generalizes the sine addition and subtraction laws. We show that each solution (f, g) such that f and g are linearly independent, satisfies the special cosine-sine functional equation <Equation ID="Equ24"> <EquationSource Format="TEX">$\begin{aligned} f(xy) = f(x)g(y) + g(x)f(y) + cf(x)f(y), \ x,y \in S, \end{aligned}$</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>y</mi> <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>c</mi> <mi>f</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> <mspace width="4pt" /> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>S</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for some <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$c \in \mathbb {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>∈</mo> <mi mathvariant="double-struck">F</mi> </mrow> </math></EquationSource> </InlineEquation>, and that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$f \circ \sigma = \beta f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∘</mo> <mi>σ</mi> <mo>=</mo> <mi>β</mi> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$g \circ \sigma \in g + \mathbb {F}f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∘</mo> <mi>σ</mi> <mo>∈</mo> <mi>g</mi> <mo>+</mo> <mi mathvariant="double-struck">F</mi> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation>.

On a sine addition/subtraction law on semigroups

期刊论文
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2025年6月11日

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Henrik Stetkær

摘要

Let S be a semigroup, $\sigma :S \rightarrow S$ $σ : S \to S$ a surjective homomorphism and $\mathbb {F}$ $F$ a field of characteristic different from 2. We study solutions $f,g :S \rightarrow \mathbb {F}$ $f, g : S \to F$ of the functional equation $\begin{aligned} f(x\sigma (y)) = f(x)g(y) + \beta g(x)f(y) + \gamma f(x)f(y), \ x,y \in S, \end{aligned}$ $\begin{matrix} f (x σ (y)) = f (x) g (y) + β g (x) f (y) + γ f (x) f (y), x, y \in S, \end{matrix}$ in which $\beta \in \mathbb {F}^*$ $β \in F^{*}$ and $\gamma \in \mathbb {F}$ $γ \in F$ are constants. The functional equation generalizes the sine addition and subtraction laws. We show that each solution (f, g) such that f and g are linearly independent, satisfies the special cosine-sine functional equation $\begin{aligned} f(xy) = f(x)g(y) + g(x)f(y) + cf(x)f(y), \ x,y \in S, \end{aligned}$ $\begin{matrix} f (x y) = f (x) g (y) + g (x) f (y) + c f (x) f (y), x, y \in S, \end{matrix}$ for some $c \in \mathbb {F}$ $c \in F$ , and that $f \circ \sigma = \beta f$ $f \circ σ = β f$ and $g \circ \sigma \in g + \mathbb {F}f$ $g \circ σ \in g + F f$ .