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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?

edited Dec 24 2010 at 17:42

Igor Rivin
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asked Dec 24 2010 at 17:39

Anthony Labarre
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An old but probably still relevant source is the book Combinatorial Identities by Riordan. – ndkrempel Dec 24 2010 at 17:42

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This question is rather general, since a lot of mathematics (eg, enumerative combinatorics, enumerative algebraic geometry, etc) can be stated in this form... – Igor Rivin Dec 24 2010 at 17:43

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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. – Gerry Myerson Dec 24 2010 at 20:31

dke · Answer 1 · 2010-12-25 18:42:48Z UTC

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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...)

answered Dec 25 2010 at 18:42

dke
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Thanks, I did not know about that reference. – Anthony Labarre Dec 26 2010 at 10:21

Anixx · Answer 2 · 2010-12-25 17:48:26Z UTC

Some (but not all) formulas and approaches:

http://en.wikipedia.org/wiki/Indefinite_sum

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

http://en.wikipedia.org/wiki/Derivatives_and_integrals_of_elementary_functions_in_alternative_calculi

Open problems and known identities involving sums

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