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Primer: Let $n \in \mathbb{N}$ and $\epsilon = \exp(i(2\pi)/n)$. Then for any polynomial $F(x) = f_0 + f_1x + f_2x^2 + \dots$, where $f_k = 0$ when $k > \deg(F)$, the sum $f_0 + f_n + f_{2n} + \dots$ is given by $f_0 + f_n + f_{2n} + \dots = \frac{1}{n}(F(1) + F(\epsilon) + \dots + F(\epsilon^{n-1}))$.

To my knowledge the sum equality above is the so-called root of unity filter, and with it we can quite easily find the sum the coefficients $f_k$ from the constructed polynomial, where $k$ is a non-negative multiple of $n$. What I'm still struggling with is how to properly pose the question when one wants to use the said filter. Namely, the name of the game seems to be to represent the desired quantity as a sum of specific index multiple coefficients from a polynomial.

As an example, we could (and should) use the filter to evaluate the sum $\sum_{k\geq 0}{n \choose k}$. But what (if any) meaning does the polynomial itself have? I mean that what answer would you give to the question: "What does this polynomial represent?" or "What do the terms of this polynomial represent?"

Epsilon Away
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    It's difficult to understand what you are trying to ask. Maybe you are looking for the notion of a generating function? – Eric Wofsey Jul 21 '21 at 16:16
  • @EricWofsey I know, it is hard for me to put it into words. The basis of my question is is that in most of my mathematical life, the given $n$th degree polynomials have had some meaning in them: This quadratic equation describes (elementary) the distance of a flying projectile, this cubic equation approximates the yield of a given factory over years. But in the case of the root of unity, we are packing the desired information into some (but not all) coefficients of some polynomial. What I am trying to do is to give meaning to this some polynomial, but I am having hard time with it. – Epsilon Away Jul 21 '21 at 16:24

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The MSE question 3213142 "Root of unity filter" has some details if you are interested. The Wikipedia article Generating function has even more details about the general concept.

Briefly, the polynomial is a means towards the end of finding the desired sum. The "trick" depends on knowing an algebraic representation of the polynomial $F(x)$ so that the right side sum in the equation $$ f_0 + f_n + f_{2n} + \dots = \frac{1}{n}(F(1) + F(\epsilon) + \dots + F(\epsilon^{n-1})) $$ can be simplified in order to find the left side sum. That is all that is required. If no simplification is possible, then the equation is still true, but useless for the purpose of finding a closed form of the sum.

In answer to your question

"What does this polynomial represent?" or "What do the terms of this polynomial represent?"

no "meaning" needs to be given to the polynomial. In some instances, depending on the context, there could be a natural meaning to the polynomial, but that would be an exceptional case. In general, the polynomial contains exactly the same amount of information as the coefficient sequence $\,\{f_0,f_1,f_2,\dots\}\,$ itself. That is, the generating function polynomial is an algebraic representation of the coefficient sequence itself.

Somos
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