A clarification in a limit While reading this answer I do not understand that how can we claim that there exist a polynomial with the properties, even if it exist then how is it same for the function and the inverse.
 A: Start with $ f ( x ) = x + a _ 2 x ^ 2 + a _ 3 x ^ 3 + a _ 4 x ^ 4 \cdots $ and $ f ^ { − 1 } ( x ) = x + A _ 2 x ^ 2 + A _ 3 x ^ 3 + A _ 4 x ^ 4 \cdots $ and substitute into $ f ^ { - 1 } ( f ( x ) ) = x $ (or into $ f ( f ^ { - 1 } ( x ) ) = x $ for the same final result):
$$ \eqalign { x & = ( x + a _ 2 x ^ 2 + a _ 3 x ^ 3 + a _ 4 x ^ 4 + \cdots ) + A _ 2 ( x + a _ 2 x ^ 2 + a _ 3 x ^ 3 + a _ 4 x ^ 4 + \cdots ) ^ 2 \\ & \quad { } + A _ 3 ( x + a _ 2 x ^ 2 + a _ 3 x ^ 3 + a _ 4 x ^ 4 + \cdots ) ^ 3 + A _ 4 ( x + a _ 2 x ^ 2 + a _ 3 x ^ 3 + a _ 4 x ^ 4 + \cdots ) ^ 4 + \cdots \\
& = x + ( a _ 2 + A _ 2 ) x ^ 2 + ( a _ 3 + 2 a _ 2 A _ 2 + A _ 3 ) x ^ 3 + ( a _ 4 + 2 a _ 3 A _ 2 + a _ 2 ^ 2 A _ 2 + 3 a _ 2 A _ 3 + A _ 4 ) x ^ 4 + \cdots \text . } $$
This holds for all $ x $ (at least sufficiently close to $ 0 $), so
$$ \displaylines { a _ 2 + A _ 2 = 0 \text , \\ a _ 3 + 2 a _ 2 A _ 2 + A _ 3 = 0 \text , \\ a _ 4 + 2 a _ 3 A _ 2 + a _ 2 ^ 2 A _ 2 + 3 a _ 2 A _ 3 + A _ 4 = 0 \text , \\ \cdots \text . } $$
We can now start solving these:
$$ \displaylines { A _ 2 = - a _ 2 \text , \\ A _ 3 = - a _ 3 - 2 a _ 2 A _ 2 = - a _ 3 + 2 a _ 2 ^ 2 \text , \\ A _ 4 = - a _ 4 - 2 a _ 3 A _ 2 - a _ 2 ^ 2 A _ 2 - 3 a _ 2 A _ 3 = - a _ 4 + 5 a _ 2 a _ 3 - 5 a _ 2 ^ 3 \text , \\ \cdots \text . } $$
So we have $ A _ n = - a _ n + P _ n ( a _ 2 , \ldots , a _ { n - 1 } ) $, where $ P _ 2 = 0 $, $ P _ 3 ( a _ 2 ) = 2 a _ 2 ^ 2 $, $ P _ 4 ( a _ 2 , a _ 3 ) = 5 a _ 2 a _ 3 - 5 a _ 2 ^ 3 $, etc.
So these polynomials exist; and it doesn't matter whether you call the coefficient sequences $ a $ and $ A $, or $ b $ and $ B $; it's the same polynomial sequence $ P $ either way.  As long as a function $ f $ is analytic at $ 0 $ with $ f ( 0 ) = 0 $ and $ f ' ( 0 ) = 1 $, then the coefficients of the function and its inverse will be related by these same polynomials.
