Is there something called the pochhammer expansion?

The general expression for $n\ge 0$ is

$$x^n=\sum_{k=0}^n{n\brace k}x^{\underline k}\,,$$

where $n\brace k$ is the Stirling number of the second kind that counts the partitions of $[n]$ into $k$ non-empty subsets. This is easily proved by induction on $n$ once you have the identity

$$x\cdot x^{\underline k}=x^{\underline{k+1}}+kx^{\underline k}$$

and the recurrence

$${n\brace k}=k{{n-1}\brace k}+{{n-1}\brace{k-1}}\,.$$

The induction step is:

$$\begin{align*} x^n&=x\sum_k{{n-1}\brace k}x^{\underline k}\\ &=\sum_k{{n-1}\brace k}x^{\underline{k+1}}+\sum_k{{n-1}\brace k}kx^{\underline k}\\ &=\sum_k{{n-1}\brace{k-1}}x^{\underline k}+\sum_k{{n-1}\brace k}kx^{\underline k}\\ &=\sum_k\left(k{{n-1}\brace k}+{{n-1}\brace{k-1}}\right)x^{\underline k}\\ &=\sum_k{n\brace k}x^{\underline k} \end{align*}$$