The symmetrized Sine addition formula

Abstract

As a continuation of An and Yang (Integral Equ Oper Theory 66:183–195, 2010) in this paper, the symmetrized Sine addition formula

$$ w(xy)+w(yx)=2f(x)w(y)+2w(x)f(y) $$

is studied, where f, w are the unknown complex functions on a group G. The structures of its general solutions are given. In particular, if there is some c in the center of G such that f(c ²) ≠ f(c)², the solutions are of the form: ${(f,w)=\left(\frac{\varphi+\psi}{2}, \lambda(\varphi-\psi)\right)}$, where φ, ψ are two distinct characters on G. The case where f is abelian is investigated in detail. In this case, f is completely determined: ${f=\frac{\varphi+\psi}{2}}$ for two characters φ, ψ on G. The properties of the corresponding w are also discussed although the full story of w still needs more efforts. The solutions (f, w) on abelian groups are also recovered from the above results. Several other cases are also considered.