Sometimes I bump into more of the astonishing results of Gosper (some examples follow) and I gather that a lot of them come from hypergeometrics and special functions.
- Have there been any attempts to try and collect together these?
- What papers/books and such are there which study collections of results like this?
$$\prod_{n=1}^{\infty} \left(\begin{matrix} -\frac{n}{2(2n+1)} & \frac{1}{2n(2n+1)} & \frac{1}{n^4} \\\\ 0 & -\frac{n}{2(2n+1)} & \frac{5}{4n^2} \\\\ 0 & 0 & 1 \end{matrix}\right) = \left(\begin{matrix} 0&0&\zeta(5)\\\\0&0&\zeta(3)\\\\0&0&1 \end{matrix}\right)$$
from A third-order Apery-like recursion for $\zeta(5)$
$$W(x)=a+\sum_{n=0}^\infty \left\{{\sum_{k=0}^n {S_1(n,k)\over \left[{\ln\left({x\over a}\right)-a}\right]^{k-1}(n-k+1)!}}\right\} \left[{1-{\ln\left({x\over a}\right)\over a}}\right]^n$$
from a page about Lambert's W-Function
$$\prod_{n=1}^{\infty} \frac{1}{e}\left(\frac{1}{3n}+1\right)^{3n+1/2}= \sqrt{\frac{\Gamma(\frac{1}{3})}{2 \pi}} \frac{3^{13/24}\exp\left(1+\frac{2\pi^2-3\psi_1\left(\frac{1}{3}\right)}{12 \pi \sqrt{3}}\right)}{A^4}$$
from Mathworld
$$\sum_{n=1}^{\infty} \frac{(-1)^2}{n^2}\cos(\sqrt{n^2 \pi^2 - 9}) = - \frac{\pi^2}{21 e^3}$$
