# Polynomial Formula like Infinite Sum with non-natural index

By polynomial formula $$(\sum_{i\in m} x_i)^n=\sum_{\substack{j_i \in \mathbb{Z}^+ \\ \sum j_i=n}}\left(\begin{array}{c} n\\ j_{0},\ldots , j_{m-1} \end{array}\right)\prod_{i \in m} x_i^{j_i}$$ where $n \in \mathbb{N}$

But what about $n \not \in \mathbb{N}$? For example $(a+b+c)^{\sqrt{3}}$? These cases probably do not have polynomial formula, but do these cases have a infinite sum like a polynomial formula?

-

## 1 Answer

When $n$ is not an integer, multivariate Taylor series expansion is to be applied. Let's expand $(a+b+c)^{\alpha}$ around $b=0$ and $c=0$. We will need: $$\frac{\partial^{p+q}}{\partial b^p \partial c^q} (a+b+c)^{\alpha} = \prod_{k=0}^{p+q-1} (\alpha -k) \cdot (a+b+c)^{\alpha-p-q} = \binom{\alpha}{p+q} (p+q)! \cdot (a+b+c)^{\alpha-p-q}$$ Thus: $$(a+b+c)^{\alpha} = \sum_{p=0}^\infty \sum_{q=0}^\infty \binom{\alpha}{p+q} (p+q)! \cdot (a)^{\alpha-p-q} \frac{b^p}{p!} \frac{c^q}{q!} = \sum_{p=0}^\infty \sum_{q=0}^\infty \binom{\alpha}{p, q, \alpha-p-q} a^{\alpha-p-q} b^p c^q$$ As you can see, in the case of $\alpha \in \mathbb{N}$, the series terminates, and the expansion reduces to the tri-nomial expansion.

-
Okay, I see, thank you. –  Popopo Sep 15 '12 at 13:56