# Gosper's Mathematics

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?

$$\prod_{n=1}^{\infty} \left(\begin{matrix} -\frac{n}{2(2n+1)} & \frac{1}{2n(2n+1)} & \frac{1}{n^4} \\\\ 0 & -\frac{n}{2(2n+1)} & \frac{5}{4n^2} \\\\ 0 & 0 & 1 \end{matrix}\right) = \left(\begin{matrix} 0&0&\zeta(5)\\\\0&0&\zeta(3)\\\\0&0&1 \end{matrix}\right)$$

$$W(x)=a+\sum_{n=0}^\infty \left\{{\sum_{k=0}^n {S_1(n,k)\over \left[{\ln\left({x\over a}\right)-a}\right]^{k-1}(n-k+1)!}}\right\} \left[{1-{\ln\left({x\over a}\right)\over a}}\right]^n$$

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

$$\sum_{n=1}^{\infty} \frac{(-1)^2}{n^2}\cos(\sqrt{n^2 \pi^2 - 9}) = - \frac{\pi^2}{21 e^3}$$

• Ramanujaniacal? – Eric Tressler Oct 5 '10 at 5:11
• SIAM publications...sort of...discuss some. – Unknown Oct 5 '10 at 7:15
• Well... there's HAKMEM: home.pipeline.com/~hbaker1/hakmem/hakmem.html – J. M. isn't a mathematician Oct 5 '10 at 15:14
• What is $A$ in the denominator of the formula for the second infinite product? – John Bentin Oct 5 '10 at 20:03
• There are two typos in the last sum. It should be $\sum_{n=1}^\infty \frac{(-1)^n}{n^2}\cos(\sqrt{n^2\pi^2-9}) = -\frac{\pi^2}{12e^3}$. – Tito Piezas III Dec 30 '12 at 14:38