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I'll jump right into the question:

Let $\omega$ be a primitive $n$th root of unity. Show that \begin{align*} \frac{1}{n} \sum_{i=0}^{n-1} \frac{1}{(1-\omega^i t)(1 - \omega^{-i}t)} = \frac{1-t^{2n}}{(1-t^2)(1-t^n)^2}. \end{align*}

I actually already know of one way of computing this. Let $G = C_n = \langle g \rangle$, the cyclic group of order $n$, and let $G$ act on $R = \mathbb{C}[x,y]$ via $g \cdot x = \omega x$, $g \cdot y = \omega^{-1} y$. Then the left hand side of the above equality is the Hilbert series of $R^G$ calculated using Molien's formula, while the right hand side is the Hilbert series for $R^G$ calculated by first observing that \begin{align*} R^G = \bigoplus_{i-0}^{n-1} \hspace{2pt} \mathbb{C}[x^n,y^n](xy)^i \end{align*} and then doing a little more work.

However, I'd like to be able to show that we have the above equality without appealing to this result. I wasn't able to evaluate the sum "by hand", unless there's a nice trick I'm missing. There might also be a more combinatorial approach using generating functions, but I haven't been able to find one.

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