As many people here, I know of a few identities involving expressions of the type $\sum_{i}\ f(i)$, with "arbitrarily complicated $f(\cdot)$", as well as closed formulas in some cases.

I also know of a few references (Concrete Mathematics, A=B) that can help me with "hard" sums. However, is there a reference on the web that lists useful identities and related open problems (i.e. expressions for which no closed forms are known)? I wouldn't expect any single reference to be comprehensive, but perhaps there are some that I should know about?

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    $\begingroup$ An old but probably still relevant source is the book Combinatorial Identities by Riordan. $\endgroup$ – ndkrempel Dec 24 '10 at 17:42
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    $\begingroup$ This question is rather general, since a lot of mathematics (eg, enumerative combinatorics, enumerative algebraic geometry, etc) can be stated in this form... $\endgroup$ – Igor Rivin Dec 24 '10 at 17:43
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    $\begingroup$ I don't know the answer to your question, but if it were my question I think I'd type "combinatorial identities" into Google and follow the URL-trail until I either found what I was looking for or convinced myself that it doesn't exist. $\endgroup$ – Gerry Myerson Dec 24 '10 at 20:31

You can find a few hundred pages of all sorts of combinatorial identities as well as summation techniques collected in the notebooks of Henry W. Gould. (Look for the 8 volumes in pdf format half way down the page...)

  • $\begingroup$ Thanks, I did not know about that reference. $\endgroup$ – Anthony Labarre Dec 26 '10 at 10:21

Some (but not all) formulas and approaches:


Some other identities and question signs where result is still open question:


  • $\begingroup$ A multiline $$x=y$$ comment. $\endgroup$ – Anton Geraschenko Jun 25 '13 at 4:56
  • $\begingroup$ @Anton Geraschenko how you do it? When I press enter a new line is not added. It works only with formulas, but not with lists, say? $\endgroup$ – Anixx Jun 25 '13 at 4:57
  • $\begingroup$ Even on the old MO, hitting enter did not affect the final displayed comment. Multiline comments were always achieved through displayed equations ... putting the math between double dollar signs. $\endgroup$ – Anton Geraschenko Jun 25 '13 at 5:00

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