# Four kinds of generalized hypergeometric formulas for $\pi$

Given,

$$\begin{array}{|c|c|c|c|} \hline n&a_n&b_n&c_n\\ \hline 1 & 6541681608 & 163096908 & -640320^3\\ \hline 2 & 85840 & 4492 & -14112^2\\ \hline 3 & 28302 & 1654 & -300^3\\ \hline 4 & 48 & 8 & -2^9\\ \hline \end{array}$$

using those $a_n,\, b_n,\, c_n$, how do we rigorously show that the corresponding four Ramanujan-type pi formulas,

\begin{aligned} \frac{1}{\pi}&=\sum_{n=0}^\infty (-1)^n \frac{(6n)!}{(3n)!n!^3}\frac{a_1n+b_1}{(-c_1)^{n+1/2}}\\ \frac{1}{\pi}&=\sum_{n=0}^\infty (-1)^n \frac{(4n)!}{n!^4}\frac{a_2n+b_2}{(-c_2)^{n+1/2}}\\ \frac{1}{\pi}&=\sum_{n=0}^\infty (-1)^n \frac{(2n)!(3n)!}{n!^5}\frac{a_3n+b_3}{(-c_3)^{n+1/2}}\\ \frac{1}{\pi}&=\sum_{n=0}^\infty (-1)^n \frac{(2n)!^3}{n!^6}\frac{a_4n+b_4}{(-c_4)^{n+1/2}} \end{aligned}

can be transformed to,

$$\pi = \frac{\sqrt{-c_1}}{b_1\Big(\,_3F_2\big(\tfrac{1}{6},\tfrac{5}{6},\tfrac{1}{2};1,1;\frac{1728}{c_1}\big)+120\frac{a_1}{b_1\,c_1} \,_3F_2\big(\tfrac{7}{6},\tfrac{11}{6},\tfrac{3}{2};2,2;\frac{1728}{c_1}\big) \Big)}$$

$$\pi = \frac{\sqrt{-c_2}}{b_2\Big(\,_3F_2\big(\tfrac{1}{4},\tfrac{3}{4},\tfrac{1}{2};1,1;\frac{256}{c_2}\big)+24\frac{a_2}{b_2\,c_2} \,_3F_2\big(\tfrac{5}{4},\tfrac{7}{4},\tfrac{3}{2};2,2;\frac{256}{c_2}\big) \Big)}$$

$$\pi = \frac{\sqrt{-c_3}}{b_3\Big(\,_3F_2\big(\tfrac{1}{3},\tfrac{2}{3},\tfrac{1}{2};1,1;\frac{108}{c_3}\big)+12\frac{a_3}{b_3\,c_3} \,_3F_2\big(\tfrac{4}{3},\tfrac{5}{3},\tfrac{3}{2};2,2;\frac{108}{c_3}\big) \Big)}$$

$$\pi = \frac{\sqrt{-c_4}}{b_4\Big(\,_3F_2\big(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2};1,1;\frac{64}{c_4}\big)+8\frac{a_4}{b_4\,c_4} \,_3F_2\big(\tfrac{3}{2},\tfrac{3}{2},\tfrac{3}{2};2,2;\frac{64}{c_4}\big) \Big)}$$

The first one I found in Mathworld's Pi Formulas, (eq.85) but was not in the form above. Hence did not transparently show its connection to the Chudnovsky algorithm.

The other generalized hypergeometric formulas I only found empirically, so it would be nice to know if someone knows of a slick way to transform it from one kind to the other.

P.S. The four $c_n$ are integer values of certain eta quotients of the Dedekind eta function, hence the "modular forms" tag.