How to use the method of undetermined coefficients to get complex version of binomial expansion? Show that if $|z|<1$, and $\alpha\in\mathbb{C}$, then $(1+z)^{\alpha}=\sum\limits_{n=0}^\infty {\alpha\choose n}z^n$, where $${\alpha\choose n}=\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}$$
I write $(1+z)^\alpha$ as $e^{\alpha\log(1+z)}$, and expand $\log (1+z)$ by Taylor expansion, and then expand $e^{\alpha\log (1+z)}$ by expansion of $e^x$. But I got stuck since I cannot think of nice way comparing the undetermined coefficients. Can some one help me out? Any hints would be appreciated!
 A: $$f(z) = (1+z)^\alpha = e^{\alpha\log(1+z)}$$
is holomorphic in the unit circle. It follows that
$$
f(z) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} \, z^n
$$
for $|z| < 1$. It is easy to verify that
$$
f^{(n)}(0) = \alpha(\alpha-1)\cdots(\alpha-n+1) \, .
$$
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*}

$$ $$

In table 1 of this paper they state besides other information following building blocks
\begin{array}{cc}
F(x)&F^\triangle(n,k)\\
\hline\\
xe^x&\frac{k^{n-k}}{(n-k)!}\\
\\
\ln(1+x)&\frac{k!}{n!}
\begin{bmatrix}
n\\
k\\
\end{bmatrix}\\
\\
e^x-1&\frac{k!}{n!}
\left\{
\begin{matrix}n\\k\end{matrix}
\right\}\\
\end{array}
with $\begin{bmatrix}
n\\
k\\
\end{bmatrix}, \left\{
\begin{matrix}n\\k\end{matrix}
\right\}$ the Stirling numbers of the first and second kind. 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
