weird series expansion

[tommy1729]

consider F(x) a real-analytic function mapping R+ to R+ with only a single fixpoint, the one at 0.

let g(x,n) be g(g(x,n-1),1) and g(x,1) = arcsinh(x/2)

F(x) is strictly nondecreasing. F(x) grows slower than x^2 + 1.

let a_n be real.

the following weird series expansion has come to my mind :

a0 + a1 x + a2 g(x,1) + a3 g(x,2) + a4 g(x,3) + ...

and many questions pop up.

regards

tommy1729

[73939]

g(x,n) = g(g(x,n-1))
The right side of the equation uses g as a function on one variable. How is g(x) defined?

[tommy1729]

i corrected it ... should be clear now.