Is there a comprehensive resource listing binomial identities? I am more interested in combinatorial proofs of such identities, but even a list without proofs will do.

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    $\begingroup$ You've seen dlmf.nist.gov/26.3 and functions.wolfram.com/GammaBetaErf/Binomial and Gradshteyn and Ryzhik and probably even the Graham-Knuth-Patashnik book I suppose? $\endgroup$ – J. M. isn't a mathematician Aug 23 '10 at 4:51
  • $\begingroup$ I am aware of Concrete Math book, but I don't have it now, hence I was looking for some online resource. I think the wolfram link you just gave is quite a good one. Thanks. $\endgroup$ – user813 Aug 23 '10 at 5:09
  • $\begingroup$ Detailing the info above you find in section 0.15 of Gradshteyn, Ryzhik, Jeffrey, Zwillinger's Table of Integrals, Series, and Products amazon.com/Table-Integrals-Products-Sixth-Gradshteyn/dp/… a list of 36 Sums of the binomial coefficients, without proof, but with a reference for it. $\endgroup$ – Américo Tavares Aug 23 '10 at 11:27
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    $\begingroup$ The pedant in me would like to add that no list will be "comprehensive" as one can generate an infinite number of binomial identities... That being said +1, I think this is a useful question. $\endgroup$ – BBischof Oct 7 '10 at 23:12
  • $\begingroup$ math.ucsd.edu/~jverstra/bijections.pdf is a link with more than a few combinatorial identities - with proofs. $\endgroup$ – NaN Dec 27 '13 at 22:46

The most comprehensive list I know of is H.W. Gould's Combinatorial Identities. It is available directly from him if you contact him. He also has some pdf documents available for download from his web site. Although he says they do "NOT replace [Combinatorial Identities] which remains in print with supplements," they still contain many more binomial identities even than in Concrete Mathematics. In general, Gould's work is a great resource for this sort of thing; he has spent much of his career collecting and proving combinatorial identities.

Added: Another useful reference is John Riordan's Combinatorial Identities. It's hard to pick one of its 250 pages at random and not find at least one binomial coefficient identity there. Unfortunately, the identities are not always organized in a way that makes it easy to find what you are looking for. Still it's a good resource.

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    $\begingroup$ Wow, thanks. Gould's site has quite a good collection. $\endgroup$ – user813 Oct 8 '10 at 16:46

If you do possess Concrete Mathematics, check out 5.1 Basic Identities and 5.2 Basic Practice. The list can be found at P174 Table 174 The top ten binomial coefficient identities.

  • $\begingroup$ I checked it out, it's good enough for what I was looking for. Thanks. $\endgroup$ – user813 Aug 24 '10 at 4:03

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