Fibonacci as iteration of fractional linear function

[Gottfried]

I found an interesting expression of the fibonacci sequence as iteration of the function
\[f(z) = \frac{1}{1+z}\]

Just to compare an exercise in iteration of mine, perhaps there is some more interesting stuff inside (Otoh I didn't arrive at a two-fixpoint analysis). It's an old material, for instance I express it in terms of "bell"-matrix (or "matrix-operator") for which I later used "carleman" matrix. Also perhaps a simple exposition of the matrix-diagonalization-concept for the fractional iteration and Schroeder-concept.