<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>Let <italic>S</italic> be a semigroup. Our main results is that we describe the complex-valued solutions of the following functional equations
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2023-0021_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable><mtr><mtd><mrow><mi>g</mi><mrow><mo>(</mo><mrow><mi>x</mi><mi>σ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>g</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mi>σ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>g</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math>
<tex-math>\matrix{   {g\left( {x\sigma \left( y \right)} \right) = g\left( x \right)g\left( y \right) + f\left( x \right)f\left( y \right),} &amp; {x,y \in S,}  \cr    {f\left( {x\sigma \left( y \right)} \right) = f\left( x \right)g\left( y \right) + f\left( y \right)g\left( x \right),} &amp; {x,y \in S,}  \cr  }</tex-math>
</alternatives>
</disp-formula>
and
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2023-0021_eq_002.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable><mtr><mtd><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mi>σ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>g</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>-</mo><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math>
<tex-math>\matrix{   {f\left( {x\sigma \left( y \right)} \right) = f\left( x \right)g\left( y \right) - f\left( y \right)g\left( x \right),} &amp; {x,y \in S,}  \cr  }</tex-math>
</alternatives>
</disp-formula>
where <italic>σ : S</italic> → <italic>S</italic> is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants. We also give some applications.</p>
</abstract>

Let S be a semigroup. Our main results is that we describe the complex-valued solutions of the following functional equations



g(xσ(y))=g(x)g(y)+f(x)f(y),x,y∈S,f(xσ(y))=f(x)g(y)+f(y)g(x),x,y∈S,
\matrix{   {g\left( {x\sigma \left( y \right)} \right) = g\left( x \right)g\left( y \right) + f\left( x \right)f\left( y \right),} & {x,y \in S,}  \cr    {f\left( {x\sigma \left( y \right)} \right) = f\left( x \right)g\left( y \right) + f\left( y \right)g\left( x \right),} & {x,y \in S,}  \cr  }


and



f(xσ(y))=f(x)g(y)-f(y)g(x),x,y∈S,
\matrix{   {f\left( {x\sigma \left( y \right)} \right) = f\left( x \right)g\left( y \right) - f\left( y \right)g\left( x \right),} & {x,y \in S,}  \cr  }


where σ : S → S is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants. We also give some applications.

Cosine and Sine Addition and Subtraction Law with an Automorphism

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"Cosine and Sine Addition and Subtraction Law with an Automorphism"}

Let S be a semigroup. Our main results is that we describe the complex-valued solutions of the following functional...

Cosine and Sine Addition and Subtraction Law with an Automorphism

Published Online: Nov 29, 2023

Page range: -

Received: Mar 26, 2023

Accepted: Nov 17, 2023

DOI: https://doi.org/10.2478/amsil-2023-0021

Keywords
functional equation, semigroup, addition law, automorphism

© 2023 Youssef Aserrar et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Cosine and Sine Addition and Subtraction Law with an Automorphism

Published Online: Nov 29, 2023

Page range: -

Received: Mar 26, 2023

Accepted: Nov 17, 2023

DOI: https://doi.org/10.2478/amsil-2023-0021

Keywordsfunctional equation, semigroup, addition law, automorphism

© 2023 Youssef Aserrar et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
functional equation, semigroup, addition law, automorphism