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

Let <italic>S</italic> be a semigroup and α, β ∈ ℝ. The purpose of this paper is to determine the general solution <italic>f</italic> : ℝ2 → <italic>S</italic> of the following parametric functional equation <disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2022-0001_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mn>1</mn></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mn>2</mn></msub><mo>+</mo><mi>α</mi><msub><mrow><mi>y</mi></mrow><mn>1</mn></msub><msub><mrow><mi>y</mi></mrow><mn>2</mn></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mn>1</mn></msub><msub><mrow><mi>y</mi></mrow><mn>2</mn></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mn>2</mn></msub><msub><mrow><mi>y</mi></mrow><mn>1</mn></msub><mo>+</mo><mi>β</mi><msub><mrow><mi>y</mi></mrow><mn>1</mn></msub><msub><mrow><mi>y</mi></mrow><mn>2</mn></msub></mrow><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mn>1</mn></msub></mrow><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mn>2</mn></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mn>2</mn></msub></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math>
<tex-math>f\left( {{x_1} + {x_2} + \alpha {y_1}{y_2},{x_1}{y_2} + {x_2}{y_1} + \beta {y_1}{y_2}} \right) = f\left( {{x_1},{y_1}} \right)f\left( {{x_2},{y_2}} \right),</tex-math>
</alternatives>
</disp-formula> for all (<italic>x</italic>1, <italic>y</italic>1), (<italic>x</italic>2, <italic>y</italic>2) ∈ ℝ2, that generalizes some functional equations arising from number theory and is connected with the characterizations of the determinant of matrices.
</abstract>

Let S be a semigroup and α, β ∈ ℝ. The purpose of this paper is to determine the general solution f : ℝ2 → S of the following parametric functional equation 


f(x1+x2+αy1y2,x1y2+x2y1+βy1y2)=f(x1,y1)f(x2,y2),
f\left( {{x_1} + {x_2} + \alpha {y_1}{y_2},{x_1}{y_2} + {x_2}{y_1} + \beta {y_1}{y_2}} \right) = f\left( {{x_1},{y_1}} \right)f\left( {{x_2},{y_2}} \right),

 for all (x1, y1), (x2, y2) ∈ ℝ2, that generalizes some functional equations arising from number theory and is connected with the characterizations of the determinant of matrices.

A Parametric Functional Equation Originating from Number Theory

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"A Parametric Functional Equation Originating from Number Theory"}

Let S be a semigroup and α, β ∈ ℝ. The purpose of this paper is to determine the general solution f : ℝ2 → S of the following parametric functional equation...

A Parametric Functional Equation Originating from Number Theory

Published Online: Jan 17, 2022

Page range: 71 - 91

Received: Jan 10, 2021

Accepted: Jan 03, 2022

DOI: https://doi.org/10.2478/amsil-2022-0001

Keywords
functional equation, number theory, character, multiplicative function, monoid, additive function

© 2022 Aziz Mouzoun et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

A Parametric Functional Equation Originating from Number Theory

Published Online: Jan 17, 2022

Page range: 71 - 91

Received: Jan 10, 2021

Accepted: Jan 03, 2022

DOI: https://doi.org/10.2478/amsil-2022-0001

Keywordsfunctional equation, number theory, character, multiplicative function, monoid, additive function

© 2022 Aziz Mouzoun et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
functional equation, number theory, character, multiplicative function, monoid, additive function