Let $\boldsymbol{x} = \left(x_1, x_2, \dots x_{D}\right)^{T}$ and consider the expression
$$P\left(\boldsymbol{x}\right) = \prod_{k=1}^{D}\left(x_k + \alpha\right)$$ where $\alpha$ is a fixed given real number.
We see that for $D = 2$ $$ P\left(\boldsymbol{x}\right) = \left(x_1+\alpha\right)\left(x_2+\alpha\right) = x_1x_2 +\alpha\left(x_1+x_2\right) + \alpha^4 $$ For $D = 3$ $$ P\left(\boldsymbol{x}\right) = \left(x_1+\alpha\right)\left(x_2+\alpha\right)\left(x_3+\alpha\right) = x_1x_2x_3 + \alpha\left(x_1x_2 + x_1x_3 + x_2x_3\right) + \alpha^2\left(x_1+x_2+x_3\right) + \alpha^3 $$ and for $D=4$ $$ \begin{align} P\left(\boldsymbol{x}\right) &= \left(x_1+\alpha\right)\left(x_2+\alpha\right)\left(x_3+\alpha\right)\left(x_4+\alpha\right) \\ &= x_1x_2x_3x_4 + \alpha\left(x_1x_2x_3 + x_1x_2x_4 + x_2x_3 x_4 + x_2x_3\right) + \alpha^2\left(x_1x_2+x_1x_3+ x_1x_4 + x_2x_3 + x_2 x_4 + x_3 x_4\right) + \alpha^3\left(x_1+x_2+x_3+x_4\right) + \alpha^4\end{align} $$ We see a kind of pattern repeating :
- The terms $x_1 x_2 \dots x_D$ and $\alpha^D$
- Each terms is a combination of $\alpha^k \times$ some products of the $x_i$'s
When we sort by factorizing by the powers of $\alpha$ we see that it is factor of some kind of permutation that I don't know how to write properly. I explain: I feel like a term looking like "$x_1x_2 + x_1x_3 + x_2x_3$" corresponds to the product of all possible permutations of $\left\{1,2,3\right\}$ but it is more complicated than that because "$x_1x_2+x_1x_3+ x_1x_4 + x_2x_3 + x_2 x_4 + x_3 x_4$" looks like a sum for all permutation but not from $\left{1,2,3,4\right\}$
My question might seem a little bit stupid but ... how can I write mathematically the pattern I am seeing ? I'm trying to write $P$ as $$ P\left(\boldsymbol{x}\right) = \sum_{k=1}^{D}\left(\text{something}\right) \cdot \alpha^k $$
