In his classic paper "Modular Equations and Approximations to $\pi$ (1914)", Ramanujan gives a standard technique to obtain a general family of series for $1/\pi$ based on series for $(2K/\pi)^{2}$ in terms of $k$ (see the details here). Most of the series which Ramanujan provides are based on the following expressions for $(2K/\pi)^{2}$:

\begin{align}\left(\frac{2K}{\pi}\right)^{2}&= 1 + \left(\frac{1}{2}\right)^{3}(2kk')^{2} + \left(\frac{1\cdot 3}{2\cdot 4}\right)^{3}(2kk')^{4} + \cdots\tag{1}\\ \left(\frac{2K}{\pi}\right)^{2}&= (1 + k^{2})^{-1}\,_{3}F_{2}\left(\frac{1}{4},\frac{3}{4},\frac{1}{2}; 1, 1; \left(\frac{g^{12} + g^{-12}}{2}\right)^{-2}\right)\tag{2}\\ \left(\frac{2K}{\pi}\right)^{2}&= (1 - 2k^{2})^{-1}\,_{3}F_{2}\left(\frac{1}{4},\frac{3}{4},\frac{1}{2}; 1, 1; -\left(\frac{G^{12} - G^{-12}}{2}\right)^{-2}\right)\tag{3}\\ \left(\frac{2K}{\pi}\right)^{2}&= \{1 - (kk')^{2}\}^{-1/2}\,_{3}F_{2}\left(\frac{1}{6},\frac{5}{6}, \frac{1}{2};1;1; \frac{27G^{24}}{(4G^{24} - 1)^{3}}\right)\tag{4}\end{align}

(In the above $G = (2kk')^{-1/12}, g = (2k/k'^{2})^{-1/12}$. I have left the series related to functions ${}_{3}F_{2}(1/3, 2/3, 1/2; 1; 1; a(k))$ based on elliptic functions to base 3).

Next Chudnovsky brothers (around 1989) use the following series $$\left(\frac{2K}{\pi}\right)^{2} = \{1 - 4G^{-24}\}^{-1/2}\,_{3}F_{2}\left(\frac{1}{6},\frac{5}{6}, \frac{1}{2};1;1; \frac{-27G^{48}}{(G^{24} - 4)^{3}}\right)\tag{5}$$ to give their famous series based on $G_{163}$. Another series can be obtained from $(5)$ above by changing the nome $q$ into $(-q)$: $$\left(\frac{2K}{\pi}\right)^{2} = \{k'^{4} + 16k^{2}\}^{-1/2}\,_{3}F_{2}\left(\frac{1}{6},\frac{5}{6}, \frac{1}{2};1;1; \frac{27g^{48}}{(g^{24} + 4)^{3}}\right)\tag{6}$$

The general family of series for $1/\pi$ based on equation $(6)$ is as follows: $$\frac{1}{\pi} = \sum_{m = 0}^{\infty}\frac{(1/6)_{m}(5/6)_{m}(1/2)_{m}}{(m!)^{3}}(A + mB)X_{n}^{m}\tag{7}$$ where \begin{align} X_{n} &= \frac{27g_{n}^{48}}{(g_{n}^{24} + 4)^{3}}\notag\\ A &= \frac{1}{2k\sqrt{g_{n}^{24} + 4}}\left(\frac{\sqrt{n}}{3}(1 - 2k^{2}) - \frac{R_{n}(k, k')}{6}\right) - \sqrt{n}\cdot\frac{g_{n}^{12}(k^{2} + 7)}{4(g_{n}^{24} + 4)^{3/2}}\notag\\ B &= \sqrt{n}(g_{n}^{12} + k)\cdot\frac{g_{n}^{24} - 8}{(g_{n}^{24} + 4)^{3/2}}\notag\\ n &> 4\notag\\ k &= k(q) = k(e^{-\pi\sqrt{n}})\notag \end{align}

I am currently trying to work out a series based on the above equation $(7)$ (using $n = 58$) and bit bogged down into calculations with various radicals. I don't know if the equation $(7)$ above has been used anywhere to generate series for $1/\pi$ (let me know of any references if this has already been in literature). I want to know whether this approach would lead a genuine new family of series for $1/\pi$.


1 Answer 1


See page 5 in http://arxiv.org/abs/1112.3259 Are the series presented there somewhat related to your series (7)?

  • 1
    $\begingroup$ This paper gives tranformations to obtain a series for $1/\pi$ from a given series for $1/\pi$. While the series given in the paper are not exactly the same as I have derived equation $(7)$, I must give +1 for the nice paper. $\endgroup$ Feb 24, 2015 at 3:33

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