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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? $\endgroup$ Aug 23, 2010 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, 2010 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. $\endgroup$ Aug 23, 2010 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, 2010 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, 2013 at 22:46

2 Answers 2

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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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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.

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