Let $p$ be an odd prime, $\zeta $ a primitive $p-$th root of unity and 
$${a\_n}(x) = \sum\limits\_{k = 1}^{p - 1} {\prod\limits\_{j = 1}^n {\left( {1 + {\zeta ^{jk}}x} \right)} } .$$ It seems that for $0 \leqslant i < p$ 
$${a\_{pn + i}}(x) = {b\_i}(x){\left( {1 + {x^p}} \right)^n}$$ 
for some polynomial $b\_i(x)$ of degree $i$.


Such things are probably well known. I would like to know where I can find references.



Simplification of a sum with roots of unity
For fixed $k$, the product of $(1+\zeta^{jk}x)$ over $p$ consecutive values of $j$ equals $1+x^p$. Your claim follows.


