How to find the polynomial expansion of $S=\prod_{i=1}^n (1+\rho ^ {i-1} \theta)$? Suppose I have a product $S=\prod_{i=1}^n (1+\rho ^ {i-1} \theta)$. How do I find a general formula for the coefficients $\alpha_i$ such that $S=\sum_{i=0}^{n} \alpha_i \theta^i$ ?
Thanks.

Being the constant term of $S$, $\alpha_0 = 1$, the product of the constants in each factor $1 + \rho^{i-1}\theta$. If $1 < k \le n$, $\alpha_k$ is the sum of the products $\rho^{i_1 -1}\rho^{i_2 - 1}\cdots \rho^{i_k - 1}$, as $1 \le i_1 < i_2 < \cdots < i_k \le n$. So \begin{align}\alpha_k &= \sum_{1 \le i_1 < i_2 <\cdots < i_k \le n} \rho^{i_1 - 1}\cdots \rho^{i_k - 1}\\
&= \sum_{0 \le i_1 - 1 < i_2 - 1 < \cdots < i_k - 1 \le n - 1} \rho^{i_1 - 1}\cdots \rho^{i_k - 1}\\
&= \sum_{0 \le j_1 < j_2 < \cdots < j_k \le n-1} \rho^{j_1 + \cdots + j_k}\\
&= \sum_{\underset{|S| = k}{S\subseteq \{0,\ldots, n-1\}}} \rho^{\sum_{j\in S} j}.
\end{align}
In particular, $$\alpha_1 = \rho^0 + \rho^1 + \cdots + \rho^{n-1} = \begin{cases}\frac{1 - \rho^n}{1 - \rho}& \rho \neq 1\\ n & \rho = 1\end{cases}$$ and $\alpha_n = \rho^{0 + 1 + \cdots + (n-1)} = \rho^{n(n-1)/2}$.
