including infinite products

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 [proof]
$\prod_{n=1}^{\infty} (1-x^n) = \sum_{n=-\infty}^{\infty}  (-1)^n  x^{\frac{n(3n-1)}{2}} $ (Euler)
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$\prod_{n=1}^{\infty} (1-x^n)^3=\sum_{n=1}^{\infty} (-1)^{n+1}(2n-1) x^{\frac{n(n-1)}{2}$
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\prod_{m=1}^{\infty}(1-x^{2m})(1+x^{2m}y)(1+x^{2m-2}y^{-1})=\sum_{n=-\infty}^{\infty}x^{n(n+1)}y^n$(Jacobi)
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$\prod_{n=1}^{\infty}{(1+x^{2n})}\prod_{n=1}^{\infty}{(1+x^{2n-1})}\prod_{n=1}^{\infty}{(1-x^{2n-1})}=1$

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$\prod_{n=2}^{\infty} (1-\frac{1}{n^2})  = \frac{1}{2}$
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$\prod_{n=2}^{\infty} (1-\frac{1}{n^3})  = \frac{\cosh({\frac{\pi\sqrt{3}}{2}})}{3\pi}$
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$\prod_{n=2}^{\infty} (1-\frac{1}{n^4})  = \frac{\sinh{\pi}}{4\pi}$
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$\prod_{n=1}^{\infty} (1+(-1)^n\frac{a}{2n+1})  = \frac{\sqrt{2}}{a+1}\sin\frac{(a+1)\pi}{2}$
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$\prod_{n=2}^{\infty} \frac{n^2-1}{n^2+1}=\frac{\pi}{\sinh(\pi)}$
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$\prod_{n=2}^{\infty} \frac{n^3-1}{n^3+1}=\frac{2}{3}$
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$\prod_{n=2}^{\infty} \frac{n^4-1}{n^4+1}=\frac{\pi\sinh{\pi}}{\cosh{(\sqrt{2}\pi)}-\cos{(\sqrt{2}\pi)}}$
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$ \prod_{n=1}^{\infty} \frac {4n^2}{4n^2-1} = \frac{\pi}{2} $ (Wallis)
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$ \prod_{n=1}^{\infty} (1- \frac{x^2}{n^2\pi^2}) =\frac{\sin x}{x} $
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$\prod_{n=1}^{\infty}\cos{\frac{x}{2^n}}=\frac{\sin{x}}{x}$
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$\prod_{n=1}^{\infty}(1-\frac{4}{3}\sin^2\frac{x}{3^n})=\frac{\sin{x}}{x}$
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$\prod_{n=1}^{\infty} (1-\frac{1}{2n^2})  = \frac{2}{\pi}$
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$\prod_{n=1}^{\infty} (1-\frac{1}{2n^2+1})  = \frac{\pi}{4}$
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$\prod_{n=1}^{\infty} (1+\frac{(-1)^{(n+1)}}{2n-1})  = \sqrt{2}$
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$\prod_{n=1}^{\infty} (1+\frac{(-1)^{(n+1)}}{2n-1})  = \sqrt{2}$
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$\prod_{n=0}^{\infty}(1+x^{2i+1})^8=\prod_{n=0}^{\infty}(1-x^{2n+1})^8+16x\prod_{n=0}^{\infty}(1-x^{2n+2})^8~~~~~~~~~~|x|<1$
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$\prod_{n=0}^{\infty}\frac{(1-axy^n)}{(1-xy^n)}=\sum_{n=0}^{\infty}\frac{\displaystyle \prod_{m=1}^{n}(1-ay^{m-1})}{\displaystyle \prod_{m=1}^{n}(1-y^m)}x^n~~~~~~~~~~|y|<1$
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$\prod_{n=1}^{\infty}\frac{1}{(1-q^{5n-1})(1-q^{5n-4})}= \sum_{n=0}^{\infty} \frac{q^{n^2}}{\displaystyle\prod_{m=1}^{n}(1-q^m)}~~~~~~~~~~|x|<1$
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$\prod_{n=1}^{\infty}\frac{1}{(1-x^{5n-2})(1-x^{5n-3})}= \sum_{n=0}^{\infty} \frac{x^{n^2+n}}{\displaystyle\prod_{m=1}^{n}(1-x^m)}~~~~~~~~~~|x|<1$