# Rapidly Converging Series for Particular Values of the Gamma Function

C. H. Brown found some rapidly converging infinite series for particular values of the gamma function

$$\frac {\Gamma\left(\tfrac 13\right)^6}{12\pi^4}=\frac 1{\sqrt{10}}\sum\limits_{k=0}^{\infty}\frac {(6k)!}{k!^3(3k)!}\frac {(-1)^k}{160^{3k}3^k}$$$$\frac {\Gamma\left(\tfrac 14\right)^4}{128\pi^3}=\frac 1{\sqrt u}\sum\limits_{k=0}^{\infty}\frac {(6k)!}{k!^3(3k)!}\frac {(2w)^k}{6486^{3k}}$$ Where\begin{align*} & u=273+180\sqrt2\\ & v=1+\sqrt{2}\\ & w=\frac {6486^3}{4u^3v^6\sqrt{2}}\end{align*}

Questions:

1. How did C. Brown derive these respective infinite formulas?
2. 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?$$j(\tau)=\frac 1q+744+196884q+21493760q^2+\cdots$$

• Looks somewhat like a BBP-style formula, the kind Ramanujan would solve in his sleep using Hypergeometric functions. I'm willing to bet the sums were discovered by a computer, and educated guess as to what such a sum might look like, and a lot of case checking until the computer found a likely match that was then formally proved – Brevan Ellefsen Jul 6 '17 at 22:25
• Reference to the Brown results please. – Somos Jul 6 '17 at 22:48
• @Somos Oh lol I just found it on Wikipedia. But all Wikipedia does is present the result, but not a proof. en.m.wikipedia.org/wiki/Particular_values_of_the_Gamma_function – Crescendo Jul 6 '17 at 22:52