A generalization of the cosine addition law on semigroups

Aequationes mathematicae - Our main result is that we describe the solutions $$g,f:S\rightarrow \mathbb {C}$$ of the functional equation $$\begin{aligned} g(x\sigma (y))=g(x)g(y)-f(x)f(y)+\alpha...     

The Cosine–Sine functional equation on a semigroup with an involutive automorphism

We determine the complex-valued solutions of the following extension of the Cosine–Sine functional equation

$$\begin{aligned} f(x\sigma (y))=f(x)g(y)+g(x)f(y)+h(x)h(y),\quad x,y\in S, \end{aligned}$$

where S is a semigroup generated by its squares and \(\sigma \) is an involutive automorphism of S. We express the solutions in terms of multiplicative and additive functions.

     

Integral Van Vleck’s and Kannappan’s functional equations on semigroups

In this paper we study the solutions of the integral Van Vleck’s functional equation for the sine

$$\begin{aligned} \int _{S}f(x\tau (y)t)d\mu (t)-\int _{S}f(xyt)d\mu (t) =2f(x)f(y),\; x,y\in S \end{aligned}$$

and the integral Kannappan’s functional equation

$$\begin{aligned} \int _{S}f(xyt)d\mu (t)+\int _{S}f(x\tau (y)t)d\mu (t) =2f(x)f(y),\; x,y\in S, \end{aligned}$$

where S is a semigroup, \(\tau \) is an involution of S and \(\mu \) is a measure that is a linear combination of Dirac measures \((\delta _{z_{i}})_{i\in I}\), such that for all \(i\in I\), \(z_{i}\) is contained in the center of S. We express the solutions of the first equation by means of multiplicative functions on S, and we prove that the solutions of the second equation are closely related to the solutions of d’Alembert’s classic functional equation with involution.

     

A New Variant of Wilson’s Functional Equation on Monoids

Acta Mathematica Sinica, English Series - We find on a monoid M the complex-valued solutions f, g : M → ? such that f is central and g is continuous of the functional equation $$f\left(...