Generalized hyperbolic functions, circulant matrices and functional equations - Muldoon, Martin E. (Linear algebra and its applications, 2005)
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). 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.