MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

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

from On some strange summation formulas

share|cite|improve this question
This looks Ramanujan-esque to me... – dvitek Oct 5 '10 at 1:42
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: – J. M. 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

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.)

share|cite|improve this answer

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.

share|cite|improve this answer

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.