Alienation of the quadratic, exponential and d’Alembert functional equations

Let \((S,+,0)\) $(S,+,0)$ be a commutative monoid, \(\sigma :S\rightarrow S\) $\sigma :S\to S$ be an endomorphism with \(\sigma ^2=id\) ${\sigma }^2=id$ and let K be a field of characteristic different from 2. We study the solutions \(f,g,h:S\rightarrow K\) $f,g,h:S\to K$ of the Pexider type functional equation \(\begin{aligned} f(x+y)+f(x+\sigma y)+g(x+y)=2f(x)+2f(y)+g(x)g(y) \end{aligned}\) $$\begin{array}{c}\hfill f(x+y)+f(x+\sigma y)+g(x+y)=2f(x)+2f(y)+g(x)g(y)\end{array}$$ resulting from summing up the well known quadratic and exponential functional equations side by side. We show that under some additional assumptions the above equation forces f and g to split back into the system of two equations \(\begin{aligned} \left\{ \begin{array}{ll}f(x+y)+f(x+\sigma y)=2f(x)+2f(y)\\ g(x+y)=g(x)g(y)\end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}f(x+y)+f(x+\sigma y)=2f(x)+2f(y)\hfill \\ g(x+y)=g(x)g(y)\hfill \end{array}\right)\end{array}$$ for all \(x,y\in S\) $x,y\in S$ (alienation phenomenon). We also consider an analogous problem for the quadratic and d’Alembert functional equations as well as for the quadratic, exponential and d’Alembert functional equations.