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 composition 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.
 A: 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.
A: Please refer this link:


*

*http://unapologetic.wordpress.com/2008/09/24/composition-of-power-series/
A: There is a closed form, but it is kind of complicated: https://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula
A: 
The paper Composita and its properties by V.V. Kruchinin and D.V. Kruchinin presents techniques to obtain the coefficients of compositae of formal power series.


They start with a given generating function $F(x)=\sum_{n\geq 1}f(n)x^n$, consider various other functions $G(x)=\sum_{n\geq 0}g(n)x^n$ and analyze the composition of $G$ with $F$.
\begin{align*}
G(F(x))&=\sum_{k\geq 0}g(k)\left[F(x)\right]^k\\
&=\sum_{k\geq 0}g(k)\sum_{n\geq k}F^\triangle(n,k)x^n
\end{align*}
with the so-called Composita $F^\triangle(n,k)$
\begin{align*}
F^\triangle(n,k)=\sum_{{\lambda_1+\cdots\lambda_k=n}\atop{
\lambda_1,\ldots,\lambda_k\geq 1}}f(\lambda_1)\cdots f(\lambda_k)
\end{align*}

Note the constant term of $F(x)$ is zero, so that the composition of generating functions is valid.
Other related papers by V.V. Kruchinin are

*

*Composition of ordinary generating functions


*Derivation of Bell Polynomials of the Second Kind
A: Here is another reference on this topic for those interested in pursuing it further.  It's in the Mathematics Magazine and so intended for undergraduate level.
MR3324698
Towse, Christopher
Iteration of sine and related power series.
Math. Mag. 87 (2014), no. 5, 338–349.
