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To prove the third identity, it suffices to take the limit $z\to -\frac12$ of the Ramanujan's cubic transformation. It yields $$_2F_1\left(\frac13,\frac23;1;-\frac18\right)=\lim_{z\to-\frac12^+} \frac{_2F_1\left(\frac13,\frac23;1;1-\left(\frac{1-z}{1+2z}\right)^3\right)}{1+2z}=\frac23\,C_{\mathrm{B4CC}}^2,\tag{1}$$ where the limit is evaluated using the ...


In short, there is no good way to determine the first occurrence of a string of digits within $\pi$. In fact, we do not know whether every finite string of digits actually occurs in $\pi$. [We believe this is true, but we do not know how to show it]. For some particular constants, it's rather easy. For instance, perhaps the simplest normal number is the ...

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