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2
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Closed form for $\sum_{n=0}^\infty\frac{\Gamma\left(n+\tfrac14\right)}{2^n\,(4n+1)^2\,n!}$
I was experimenting with hypergeometric-like series and discovered the following conjecture (so far confirmed by more than $5000$ decimal digits):
$$\sum_{n=0}^\infty\frac{\Gamma\!\left(n+\tfrac14\...
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26
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Integral ${\large\int}_0^1\ln^3\!\left(1+x+x^2\right)dx$
I'm interested in this integral:
$$I=\int_0^1\ln^3\!\left(1+x+x^2\right)dx.\tag1$$
Can we prove that
$$\begin{align}I&\stackrel{\color{gray}?}=\frac32\ln^33-9\ln^23+36\ln3+2\pi^2\ln3-\frac{4\pi^2}...
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