Here's an easy introduction to the basics, "Pi Formulas and the Monster Group".
http://sites.google.com/site/tpiezas/0013
Update: Just to make this more intriguing, define the fundamental unit $U_{29} = \frac{5+\sqrt{29}}{2}$ and fundamental solutions to Pell equations,
$$\big(U_{29}\big)^3=70+13\sqrt{29},\quad \text{thus}\;\;\color{blue}{70}^2-29\cdot\color{blue}{13}^2=-1$$
$$\big(U_{29}\big)^6=9801+1820\sqrt{29},\quad \text{thus}\;\;\color{blue}{9801}^2-29\cdot1820^2=1$$
$$2^6\left(\big(U_{29}\big)^6+\big(U_{29}\big)^{-6}\right)^2 =\color{blue}{396^4}$$
then we can see those integers all over the formula as,
$$\frac{1}{\pi} =\frac{2 \sqrt 2}{\color{blue}{9801}} \sum_{k=0}^\infty \frac{(4k)!}{k!^4} \frac{29\cdot\color{blue}{70\cdot13}\,k+1103}{\color{blue}{(396^4)}^k}$$
See also this MO post.
What follows is taken directly from Borweins' Pi and the AGM.
Let $N$ be a positive number and $q_N=e^{-\pi\sqrt{N}}$ and $$k_N=\frac{\vartheta_{2}^{2}(q_N)}{\vartheta_{3}^{2}(q_N)},k'_N=\sqrt{1-k_N^2},G_N=(2k_Nk'_N)^{-1/12},g_N=\left(\frac{2k_N}{{k'} _N^{2}}\right)^{-1/12}\tag{1}$$ where $\vartheta _2,\vartheta_3$ are theta functions of Jacobi defined by $$\vartheta_{2}(q)=\sum_{n\in\mathbb {Z}} q^{(n+(1/2))^2},\, \vartheta_{3}(q)=\sum_{n\in\mathbb {Z}} q^{n^2}\tag{2}$$ Borwein brothers define another variable $$\alpha(N) =\frac{E(k'_N)} {K(k_N)} - \frac{\pi} {4K^2(k_N)}\tag{3}$$ where $K, E$ denote standard elliptic integrals $$K(k) =\int_{0}^{\pi/2}\frac{dx}{\sqrt{1-k^2\sin^2x}},\,E(k)=\int_{0}^{\pi/2}\sqrt{1-k^2\sin^2x}\,dx\tag{4}$$ It is well known that $k_N, k'_N, \alpha(N) $ are algebraic when $N$ is a positive rational number. We assume $N$ to be a positive integer unless otherwise stated.
Borweins present two class of series for $1/\pi$ based on Ramanujan's ideas which are of interest here: $$\frac{1}{\pi}=\sum_{n=0}^{\infty} \dfrac{\left(\dfrac{1}{4}\right)_n\left(\dfrac{2}{4}\right)_n\left(\dfrac{3}{4}\right)_n} {(n!)^{3}}d_n(N) x_N^{2n+1}\tag{5}$$ where $$x_N=\frac{2} {g_N^{12}+g_N^{-12}} =\frac{4k_N{k'} _{N}^{2}}{(1+k_N^2)^2}, \\ d_n(N) =\left(\frac{\alpha(N) x_N^{-1}}{1+k_N^2}+\frac{\sqrt{N} }{4}g_N^{-12}\right) +n\sqrt{N} \left(\frac{g_N^{12}-g_N^{-12}}{2}\right) \tag{6}$$ and $$\frac{1}{\pi}=\sum_{n=0}^{\infty} (-1)^n\dfrac{\left(\dfrac{1}{4}\right)_n\left(\dfrac{2}{4}\right)_n\left(\dfrac{3}{4}\right)_n} {(n!)^{3}}e_n(N) y_N^{2n+1} \tag{7}$$ where $$y_N=\frac{2} {G_N^{12}-G_N^{-12}} =\frac{4k_Nk'_{N}}{1-(2k_Nk'_N)^2}, \\ e_n(N) =\left(\frac{\alpha(N) y_N^{-1}}{{k'} _{N} ^{2}-k_N^2}+\frac{\sqrt{N} }{2}k_N^2G_N^{12}\right) +n\sqrt{N} \left(\frac{G_N^{12}+G_N^{-12}}{2}\right)\tag{8}$$ The series in question is based on $(5)$ with $N=58$. Another (not so famous but equally remarkable) series given by Ramanujan using $(7)$ with $N=37$ is as follows: $$\frac{4}{\pi}=\frac{1123}{882}-\frac{22583}{882^3}\cdot\frac{1}{2}\cdot\frac{1\cdot 3}{4^2}+\frac{44043} {882^5}\cdot\frac{1\cdot 3}{2\cdot 4}\cdot\frac{1\cdot 3\cdot 5\cdot 7}{4^2\cdot 8^2}-\dots\tag{9}$$ Borweins mention that the values of $\alpha(37),\alpha(58)$ (leading to $1123$ in series $(9)$ and $1103$ in series in question) were obtained by calculating $e_0(37)$ and $d_0(58)$ to high precision.
The details of these calculations are not revealed by Borwein Brothers. But it appears that using value of $\pi$ given by Gosper and the series in question (as well as series $(9)$) one can get the values of $d_0(58),e_0(37)$ to high precision. Further some amount of computation is needed to get the minimal polynomials for $\alpha(37),\alpha(58)$. And then one can get their values in closed form as radicals.
The procedure is similar to what you have done in one of your papers dealing with evaluation of coefficients in Chudnovsky formula but the analysis is probably more complicated because the functions involved here are not like the $j$ invariant taking integer values.
To sum up, Gosper role was important in this proof on two fronts. First is the computation itself and second is that his computation brought Ramanujan's formula into limelight. It was sitting there in his paper Modular equations and approximations to $\pi$ since 1914 and no one before Gosper even looked at it.
Also as most references indicate Gosper's computation was done earlier without any knowledge of Borwein's work which got published later. So Gosper was not aware of the proof of the formula and didn't know that his computation would someday be used as a part of the proof of the series which he computed.
Best Answer
I know this general result: $$\displaystyle \sum_{k=1}^{\infty}\frac{\cos\bigl({k\pi-\sqrt{k^2\pi^2-a^2}\,}\bigr)}{k^2}=\frac{\pi^2}{12}\,\bigl({-\cosh(a)+\frac{3}{a}\sinh(a)}\bigr) $$
and you can see this paper:On some strange summation formulas by R. William Gosper 1993 can download by:http://projecteuclid.org/euclid.ijm/1255987146