The author offers as main result the following. The general solution \(f_ j:\mathbb{C}\to\mathbb{C}\) (\(j=0,1,\dots,n-1\)) of the system of functional equations \[ f_ j(x+\omega^ m y)=\sum_{k=0}^ j\omega^{km}f_{j-k}(x)f_ k(y)+\sum_{k=j+1}^{n-1} \omega^{km}f_{n+j-k}(x)f_ k(y)\tag{j} \] (\(j=0,1,\dots,n-1\); \(x,y\in\mathbb{C}\)), where \(\omega\) is an \(n\)-th root of unity and \(m\) a fixed integer between 1 and \(n\), is given by \[ f_ j(x)={1\over n}\sum_{\ell=0}^{d-1} \sum_{k=0}^{(n/d)-1} \omega^{j(dk+\ell)} \gamma_ \ell(\omega^{-kd}x) \] (\(j=0,1,\dots,n- 1\); \(x\in\mathbb{C}\)), where \(d\) is the greatest common divisor of \(m\) and \(n\) and \(\gamma_ \ell:\mathbb{C}\to\mathbb{C}\) (\(\ell=0,1,\dots,d-1\)) are arbitrary ``generalized exponentials'', that is, arbitrary solutions of \(\gamma(x+y)=\gamma(x)\gamma(y)\) (\(x,y\in\mathbb{C}\)). \{We have corrected here a minor misprint and slightly simplified the author's formula \((H_ t)\).\} Fundamental to problem and solution is the observation that all generalized exponentials are sums of \(f_{j0}:\mathbb{C}\to\mathbb{C}\) which are solutions of \((j)\) and of \(f_{j0}(\omega x)=\omega^ j f_{j0}(x)\) (\(j=0,1,\dots,n-1\)). In the case \(n=2\) the system \((j)\) reduces for \(\omega=1\), \(m=2\) to the addition formulas of cosh and sinh: \(f_ 0(x+y)=f_ 0(x)f_ 0(y)+f_ 1(x)f_ 1(y)\), \(f_ 1(x+y)=f_ 1(x)f_ 0(y)+f_ 0(x)f_ 1(y)\) and the general solution is indeed given by \(f_ 0(x)={1\over2}(\gamma_ 0(x)+\gamma_ 1(x))\), \(f_ 1(x)={1\over2}(\gamma_ 0(x)-\gamma_ 1(x))\) with arbitrary solutions \(\gamma_ 0,\gamma_ 1\) of \(\gamma(x+y)=\gamma(x)\gamma(y)\) and also \(\gamma_ 0(x)=f_ 0(x)+f_ 1(x)\), \(\gamma_ 1(x)=f_ 0(x)-f_ 1(x)\); \(f_{00}(-x)=f_{00}(x)\), \(f_{10}(-x)=-f_{10}(x)\) for \(f_{00}(x)=\cosh ax\), \(f_{10}=\sinh ax\) (\(a\in\mathbb{C}\) arbitrary). A remark by L. Székelyhidi is included which makes it possible to generalize the result to complex valued functions on an abelian group if \(x\mapsto\omega x\) is replaced by an automorphism whose \(n\)-th power is the identity on that group.