Generalized hyperbolic functions, circulant matrices and functional equations
Abstract
There is a contrast between the two sets of functional equations f0(x+y) = f0(x)f0(y) + f1(x)f1(y), f1(x+y) = f1(x)f0(y) + f0(x)f1(y), and f0(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). 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.
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