Generalized hyperbolic functions, circulant matrices and functional equations

There is a contrast between the two sets of functional equationsf0(x+y)=f0(x)f0(y)+f1(x)f1(y),f1(x+y)=f1(x)f0(y)+f0(x)f1(y),andf0(x-y)=f0(x)f0(y)-f1(x)f1(y),f1(x-y)=f1(x)f0(y)-f0(x)f1(y)satisfied by the even and odd components of a solution of f(x + y) = f(x)f(y). Schwaiger and, later, Förg-Rob and Schwaiger considered the extension of these ideas to the case where f is sum of n components. Here we shorten and simplify the statements and proofs of some of these results by a more systematic use of matrix notation.