# Questions tagged [products]

For questions about the evaluation of finite products, or their properties. For infinite ones, use "infinite-products" tag.

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### The Product of the Roots of the Minimal Polynomial Representation for Sin$\big(\frac{\pi}{x}\big)$ for Rational x, Closed Form Needed

Here is a function that I have had fun with: if you type in solve minimalpolynomial[sin[pi/x]]=0 into Wolfram Alpha and type in x as a number, otherwise it will not know how to interpret your input, ...
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### Is my direct proof correct for $\prod_{i=2}^{n} \left(1- \frac{1}{i^2}\right) =\frac {n+1}{2n}$ good/correct? [duplicate]

I need to show via direct proof that: $$\prod_{i=2}^{n} \left (1- \frac{1}{i^2}\right) =\frac {n+1}{2n}$$ We first note that $$1-\frac{1}{i^2} = \frac{(i-1)(i+1)}{i\cdot i}.$$ Then \begin{align} \...
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### Proving that $f$ verifies $f(1−x) + f(x) = 1$

I have a function which is defined for $x \in (0,1)$ and for $p>1$ with the expression \begin{align*} f(x) = \sum_{k=0}^{p-1} \frac{(-1)^{p+k}}{(p-1-k)!(p+k)!}\left(\prod_{i=k-p+1, i\neq0}^{k+p} (...
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### Proving $\prod_{k=0}^{n-1} (n^2-k^2) = \frac{(2n)!}{2}$

I'm probably having a brain-fart but I can't figure out why this identity holds: $$\prod_{k=0}^{n-1} (n^2-k^2) = \frac{(2n)!}{2}$$ I tried using various formulae involving $\binom{2n}{n}$, without ...
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### Random Variable Multiplied with Independent Random Vector - Derivation of Product Distribution

Let $x$ be a random variable and $\mathbf{Y}$ be a random vector. Is there a general formula to derive the probability density function of $$\mathbf{Z} = \frac{1}{\sqrt{x}} \mathbf{Y} \; ?$$ I could ...
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### Efficient way of polynomial multiplication with dependency in Bipartite graph form

We use a Bipartite graph: $G=(U,V,E)$, with vertices: $U$ and $V$ to represent some task dependency. Every $U$ is a unique task, and every $V$ is a degree one polynomial, e.g. $(1+v_ix)$. Our goal is ...
I am working slowly through "Riemann's zeta function", HM Edwards, Dover Publications, 1974. At the top of page 18, I read (I have altered the notation from $p(s)$ to $P(x)$) "any ...