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clathratus

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.

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