Functional super-iteration and hierarchy of functional hyper-iterations

[andydude]

how the **** did you arrive at those coefficients ?

Regular iteration. Whenever a function \( f(x) \) has a fixed point at zero, then its iterates can be described by polynomials or geometric series. For example, if \( f(x) = f_1x + f_2x^2 + f_3x^3 + \cdots \) then \( f(0) = 0 \) is a fixed point of that function. The iterates of \( f(x) \) will then be:

• \( f^2(x) = f_1(f_1x + f_2x^2 + \cdots) + f_2(f_1x + f_2x^2 + \cdots)^2 + \cdots = f_1^2 x + (f_1 + f_1^2)f_2x^2 + \cdots \)
  
• \( f^3(x) = f_1(f_1(f_1x + f_2x^2 + \cdots) + \cdots) + \cdots = f_1^3 x + (f_1^2 + f_1^3 + f_1^4)f_2x^2 + \cdots \)
  

For \( f_1 \ne 1 \) this can be generalized to
\( f^t(x) = f_1^tx + \frac{f_1^{t-1}(f_1^{t} - 1)f_2}{f_1-1}x^2 + \cdots \)
and substituing \( t=g(x) \) in this formula, then expanding the series about x again will give the formulas in my last post.

Andrew Robbins