composition of power series

Does anyone know how to derive a formula for the coefficients.

That is if, $f(x)=\sum _{n=0}^{\infty } a_nx^n$ and $g(x)=\sum _{n=0}^{\infty } b_nx^n$

suppose the compostion is an analytic function, $h(x)=f(g(x))=\sum _{n=0}^{\infty } c_nx^n$

Is there an expression we can find for the coefficients $c_n$ in terms of $a_n$ and $b_n$? Can someone show me how its derived. I know we could substitute $g$ into $f$ and collect powers of $x$. But I believe a formula for general n may be written down.

There are (rather unwieldly) "closed-forms" in terms of Bell polynomials and other closely related combinatorial objects. However, if you are really interested in efficiently calculating compositions of power series then there are better algorithms, dating back at least to the work of Brent and Kung, from which you can find links to recent work in this area.

Please refer this link:

• http://unapologetic.wordpress.com/2008/09/24/composition-of-power-series/

There is a closed form, but it is kind of complicated: http://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula

So, today I was looking into the exact same thing. What I gathered from the above internet references seemed to imply nothing for when $g(x)$ has a constant term. This article indicates that this is an open problem in the theory and lays out conditions for when the composition is defined when $g(x)$ has a constant term. However, it offers no formula for the coefficient structure of the composition.

Does anyone have experience/ideas on how to adapt the Faà di Bruno result

$$f(g(x)) = \sum_{n=1}^\infty {\sum_{k=1}^{n} a_k B_{n,k}(b_1,\dots,b_{n-k+1}) \over n!} x^n$$

where $B_{n,k}$ are Bell polynomials, to accommodate this constant term?