Consider $a_{1} = \alpha^{N-n}, a_{2} = \alpha^{N-n+1}, a_{3} = \alpha^{N-n+2}, a_{4} = \alpha^{N-n+3}, \cdots, a_{n} = \alpha^{N-1}$ where $\alpha$ is a complex $N$th root of unity where $N = 2 + (n-1)2^{n+1}$.

Let:

$P_{1}(\alpha) = a_{1}a_{2}a_{3} \cdots a_{n}$

$P_{2}(\alpha) = (a_{1} + a_{2})(a_{1} + a_{3})\cdots(a_{n-1} + a_{n})$

$P_{3}(\alpha) = (a_{1} + a_{2} + a_{3})(a_{1} + a_{2} + a_{4})\cdots(a_{n-2} + a_{n-1} + a_{n})$

$\vdots$

$P_{n}(\alpha) = (a_{1} + a_{2} + a_{3} + \cdots + a_{n})$

Is there an elementary expression for $\Pi(\alpha) = P_{1}(\alpha)P_{2}(\alpha) \cdots P_{n}(\alpha)$?

Or given $\Pi(\alpha)$, can one say anything about $\Pi(\alpha^{r})$ where $2 \le r \le N-1$?