Tuesday, 16 September 2014

summation - Can we generalise this double sum identity?

In one of my old (Dutch) algebra/analysis problem books, I came across the following "cute" double sum identity:



nk=1kh=1h23h+1h!=n1+1n!,



which the reader was asked to prove true for all nN. The proof is relatively straightforward, with induction on n. (Although I'd love to see a more imaginative proof!)



My question is, can we somehow generalise this identity? Because this is a rather non-obvious result, I have a hunch that the textbook writer derived it from some more general theorem. But I have no idea what to look for.



Any insights will be warmly appreciated!

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