Well, to not to lose all of the impules that this fiddling gave I'll add some more comments.
a) the example function. I considered the next iterate of f(x)=-1/16+x+x^2 f2(x) = x^4 + 2*x^3 + 15/8*x^2 + 7/8*x - 31/256
as base function; this can be rewritten in fixpoint notation
f2(x) = x + (x-1/4)(x+1/4)(x -1 + sqrt(15/16)*I)(x -1 - sqrt(15/16)*I)
So f2 has four fixpoints, two of them are complex. And to investigate the half-iterate of this would allow to shift by 1/4,1/2, -1+sqrt(15/16)*I, 31/16 and compare the repective solutions.
I didn't do this yet, but would be interested in a discussion of this anyway.
b) the decremented iterated exponential f(x) = b^x-1 , f°h(x) = f°(h-1)(f(x)).
From the reformulation as function of h, where b and x are assumed to be constant,
 = a_{1_{x,u}}*u^h + a_{2_{x,u}}*u^{2h} + a_{3_{x,u}}*u^{3h} + ... )
(where u=log(b) and the a_k are functions of x and u, resp are constant for given x and u)
gives some ideas concerning moduli and cyclotomic functions; however since the coefficients a are non-integer (surely irrational) I don't see yet serious applications for modulus-considerations.
However, for cyclotomic h (abs(h)=1, arg(h)/Pi= rational) this looks, as if the powertowers) )
of cyclotomic heights are periodic with k. Also series of such powertowers should be interesting.
Also I'm trying to see, whether this description gives new insights for the inconsistency of the matrix-approach and the serial summation for the alternating tetra-series (of increasing negative heights), but I don't see this so far.
Gottfried
a) the example function. I considered the next iterate of f(x)=-1/16+x+x^2 f2(x) = x^4 + 2*x^3 + 15/8*x^2 + 7/8*x - 31/256
as base function; this can be rewritten in fixpoint notation
f2(x) = x + (x-1/4)(x+1/4)(x -1 + sqrt(15/16)*I)(x -1 - sqrt(15/16)*I)
So f2 has four fixpoints, two of them are complex. And to investigate the half-iterate of this would allow to shift by 1/4,1/2, -1+sqrt(15/16)*I, 31/16 and compare the repective solutions.
I didn't do this yet, but would be interested in a discussion of this anyway.
b) the decremented iterated exponential f(x) = b^x-1 , f°h(x) = f°(h-1)(f(x)).
From the reformulation as function of h, where b and x are assumed to be constant,
(where u=log(b) and the a_k are functions of x and u, resp are constant for given x and u)
gives some ideas concerning moduli and cyclotomic functions; however since the coefficients a are non-integer (surely irrational) I don't see yet serious applications for modulus-considerations.
However, for cyclotomic h (abs(h)=1, arg(h)/Pi= rational) this looks, as if the powertowers
Also I'm trying to see, whether this description gives new insights for the inconsistency of the matrix-approach and the serial summation for the alternating tetra-series (of increasing negative heights), but I don't see this so far.
Gottfried
Gottfried Helms, Kassel