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 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.