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I'm not sure, but I think that the function


cannot be written through more elementary functions. Am I right? I am asking this as a non-mathematician.

Or in other way: Can above expression be treated as coefficient of $z^m$ in the product

$$ (1+xz)^n\,(1+z)^m.$$ say? (This has been the one answer on the related question Binomial series with two binomial coefficents)

I hope that I do not break any rule with posting like this.....

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Relevant link: wolframalpha.com/input/?i=2F1%28-n%2C-m%3B1%3Bx%29 –  wj32 Oct 20 '12 at 22:33
Wj32, thanks a lot. Very useful link. :) –  Moki Oct 20 '12 at 23:02

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