## 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

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

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This looks Ramanujan-esque to me... – drvitek Oct 5 2010 at 1:42
Ramanujaniacal? – Eric Tressler Oct 5 2010 at 5:11
SIAM publications...sort of...discuss some. – To be cont'd Oct 5 2010 at 7:15
Well... there's HAKMEM: home.pipeline.com/~hbaker1/hakmem/hakmem.html – J. M. Oct 5 2010 at 15:14
What is $A$ in the denominator of the formula for the second infinite product? – John Bentin Oct 5 2010 at 20:03

The book 'Concrete Mathematics' by Graham, Knuth, and Patashnik discusses some of the work of Gosper, in particular a lot of detail on the Gosper-Zeilberger algorithm. (It's also a really great book generally.)

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The obvious answer to your question is to search for papers authored by Gosper, or papers with Gosper's name in the title, such as Pages from the computer files of R. William Gosper. But surely this answer is so obvious that you already know it.

A good starting point for understanding modern methods for handling hypergeometric and other special function identities (as opposed to just a list of spectacular identities) is the famous book A=B.

Many such identities begin life by being discovered experimentally and then proved afterwards. A good entry point into the experimental mathematics literature is Bailey and Borwein's website.

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Some of Gosper's results are studied, and proved, in the article "Pages from the Computer Files of R. William Gosper," by Mourad E. H. Ismail, Yu Takeuchi and Ruiming Zhang (Proceedings of the American Mathematical Society, Volume 119, Number 3, November 1993). Gosper's work is also discussed in Wolfram Koepf's book Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. See also Wikipedia's list of hypergeometric identities.

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