C. H. Brown found some rapidly converging infinite series for particular values of the gamma function , $$\frac {\Gamma\left(\tfrac 13\right)^6}{12\pi^4}=\frac 1{\sqrt{10}}\sum\limits_{k=0}^{\infty}\frac {(6k)!}{k!^3(3k)!}\frac {(-1)^k}{(3\cdot160^3)^k}$$$$\frac {\Gamma\left(\tfrac 14\right)^4}{128\pi^3}=\frac 1{\sqrt u}\sum\limits_{k=0}^{\infty}\frac {(6k)!}{k!^3(3k)!}\frac {1}{\big(2\sqrt2\,u^3v^6\big)^k}$$
where $u=273+180\sqrt2,$ and $v=1+\sqrt2$.
Questions:
- How did C. Brown derive these respective infinite formulas?
- Can you derive similar formulas for different gamma functions?
I couldn't help but notice that these formulas share a similarity to the Chudnovsky Algorithm$$\frac 1\pi=12\sum\limits_{k=0}^{\infty}\frac {(6k)!}{k!^3(3k)!}\frac {545140134k+13591409}{640320^{k+1/2}}$$So perhaps Brown used the J-function and modular forms to derive the values?$$j(\tau)=\frac 1q+744+196884q+21493760q^2+\cdots$$
That Wikipedia link points to http://www.iamned.com/math/ which has a large number of amazing formulae.
That, in turn, points to http://iamned.com/math/infiniteseries.pdf titled "An Algorithm for the Derivation of Rapidly Converging Infinite Series for Universal Mathematical Constants".