An extension of Van Vleck’s functional equation for the sine

Abstract

In [7] H. Stetkær obtained the solutions of Van Vleck’s functional equation

$$ f (x\tau(y)z_0) - f(xyz_0) = 2f(x)f(y), \quad x, y \in S $$

for the sine where S is a semigroup, ${\tau}$ is an involution of S and z ₀ is a fixed element in the center of S. The purpose of this paper is to determine the complex-valued solutions of the following extension of Van Vleck’s functional equation for the sine

$$ \mu(y)f (x\tau(y)z_0) - f(xyz_0) = 2f(x)f(y),\quad x, y \in S$$

where ${\mu : S\to \mathbb{C}}$ is a multiplicative function such that ${\mu (x\tau(x))=1}$ for all ${{x\in S}}$. Furthermore, we obtain the solutions of a variant of Van Vleck’s functional equation

$$ \mu(y)f (\sigma(y)xz_0) - f(xyz_0) = 2f(x)f(y), \quad x, y \in M $$

for the sine on a monoid M, where ${\sigma}$ is an involutive automorphism of M.