A Parametric Functional Equation Originating from Number Theory

Mouzoun, Aziz; Zeglami, Driss; Aissi, Youssef

doi:10.2478/amsil-2022-0001

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 $f (x_{1} + x_{2} + α y_{1} y_{2}, x_{1} y_{2} + x_{2} y_{1} + β y_{1} y_{2}) = f (x_{1}, y_{1}) f (x_{2}, y_{2}),$ 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.

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A Parametric Functional Equation Originating from Number Theory

Abstract

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