Generalized hyperbolic functions, circulant matrices and functional equations

There is a contrast between the two sets of functional equations 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 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) satisfied by the even and odd components of a solution of f(x+y) = f(x) f(y). J. Schwaiger and, later, W. F\"org-Rob and J. 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.