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(x1x2+αy1y2,x1y2+x2y1+βy1y2) = f(x1,y1)f(x2,y2),
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.
functional equation; number theory; character; multiplicative function; monoid; additive function
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