On a functional equation arising from number theory

This work aims to determine the general solution f:F2→Kf:F2→K of the functional equation f(ϕ(x,y,u,v))=f(x,y)f(u,v) for suitable conditions on the function ϕ:F4→F2ϕ:F4→F2, where FF will denote either RR or CC, and KK is an abelian group. Using this result, we determine the solution f:C2→C⋆f:C2→C⋆ of the functional equation f(ux−vy,uy+v(x+y))=f(x,y)f(u,v) for all x,y,u,v∈Cx,y,u,v∈C without assuming any regularity condition. Here (C⋆,⋅)(C⋆,⋅) is the multiplicative group of nonzero complex numbers.