# Tagged Questions

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### Product of Chebyshev polynomials of the second kind?

So Wikipedia has this formula for a product of two Chebyshev polynomials of the second kind evaluated at a fixed $x$ with different indices: $$U_n(x)U_m(x)=\sum_{k=o}^{n}U_{m-n+2k}(x)$$ Which would ...
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### A polynomial equality problem

$a_1,a_2,a_3,\ldots,a_n,a_{n+1}$ are fixed real numbers in $(-1,\infty)$. $x_1$ and $x_2$ are fixed real numbers in $(0,1)$. Is it possible to prove that there exists or doesn't exists a real number ...
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### Showing $\sin{\frac{\pi}{13}} \cdot \sin{\frac{2\pi}{13}} \cdot \sin{\frac{3\pi}{13}} \cdots \sin{\frac{6\pi}{13}} = \frac{\sqrt{13}}{64}$

I would like to show that $$\sin{\frac{\pi}{13}} \cdot \sin{\frac{2\pi}{13}} \cdot \sin{\frac{3\pi}{13}} \cdots \sin{\frac{6\pi}{13}} = \frac{\sqrt{13}}{64}$$ I've been working on this for a few ...
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### Coefficients of $(x-1)(x-2)\cdots(x-k)$

I'm interested in the coefficients of $x$ in the expansion of, $$(x-1)(x-2)\cdots(x-k) = x^k + P_1(k) x^{k-1} + P_2(k)x^{k-2} + \cdots + P_k(k),$$ Where $k$ is an integer. In particular I am ...
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### Proof of “factorization of polynomials” using only Complex Analysis

I ask for the proof of the following: If $p$ is a polynomial with degree $n\ge 1$ and zeros in $A\subseteq \mathbb C$ whose order (multiplicity) is given by $n:A\to \mathbb N^*$ then $A$ is finite ...
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### Prove $\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k}$ and more

The current issue (vol. 120, no. 6) of the American Mathematical Monthly has a proof by probabilistic means that $$\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k}$$ for ...
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### How was Euler able to create an infinite product for sinc by using its roots?

In the Wikipedia page for the Basel problem, it says that Euler, in his proof, found that \begin{align*} \frac{\sin(x)}{x} &= \left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 ...
### Closed form for $\prod_{1 \leq i < j \leq k} (j - i)$?
Is there a closed form for $\prod_{1 \leq i < j \leq k} (j - i)$? It looks like something like a determinant of a Vandermonde matrix, but I can't seem to get it to fit.
I have 2 polynomials $p_1(x_1,\ldots,x_n)$ and $p_2(x_1,\ldots,x_n)$, of which I have to compute the product, with a special property: The exponent of each variable is always either $0$ or $1$, where ...