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clathratus

Apparently this user prefers to keep an air of cool dog about them.

My name is John. I'm a high-school student in California. I really like integrals, especially ones that are easy to generalize. I am also a fan of series and infinite products. I enjoy working with other people and seeing new ideas, so feel free to email me at joverton2020@gmail.com if you would like to collaborate. :)

A favorite identity of mine: $$\prod_{n=1}^{\infty}(en)^{9/10}\frac{E_n(\frac12)E_n(\frac7{12})E_n(\frac1{20})E_n(\frac{13}{20})}{E_n(\frac14)E_n(\frac1{12})E_n(\frac3{20})E_n(\frac{11}{20})}\\ =\exp\left[\frac{47\mathrm G}{30\pi}+\frac34\right]\sqrt{\frac{33}{91\pi}\sqrt{\frac2\pi\frac{\sqrt[5]{11}}{\sqrt[3]{7}}\sqrt[5]{\frac{3^3}{13^{3}}}}}$$ Where $$E_n(x)=\frac{(n+x)^{n+x}}{(n-x)^{n-x}}$$ and $\mathrm G$ is Catalan's constant. I discovered this identity while investigating the function $$\mathrm{L}(x)=\frac1\pi\int_0^{\pi x}\ln(\sin t)dt$$ See here for details.

Another favorite identity of mine is $$\prod_{k\ge 1}e^{(-1)^k}\left(\frac{2k-1}{2k+1}\right)^{(-1)^k k}=\exp\left[\frac{2}{\pi}\mathrm{G}-\frac{1}{2}\right]\ .$$ Which is found using the identities $$\int_0^{1/2}\Gamma(1+x)\Gamma(1-x)dx=\frac{2}{\pi}\mathrm{G}$$ and $$\Gamma(1+x)\Gamma(1-x)=\frac{\pi x}{\sin\pi x}=1+2\sum_{k\ge1}(-1)^k\frac{x^2}{x^2-k^2}\ .$$

Note to self.

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