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We set (as usual) $\displaystyle{x \choose k} := \frac{x \cdot (x-1) \cdots (x-k+1)}{k!}$ for $x\in \mathbb{C}$.

Now we can define a function

$\displaystyle f(x) := \sum\limits_{k=0}^\infty {x \choose k}$.

Does anybody know how this function is called (I need its name, so that I can get more information about it)? I believe, it should be well-known, but I don't know its name.

Note that if we defined $\displaystyle{x \choose k} := \frac{x^k}{k!}$ instead, we would simply get the $\exp$ function - so my function is probably be related to it.

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    Another good hint (in addition to that of @anon) is the fact that $f(n) = 2^n$ for every nonnegative integer $n$.2012-01-23

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By Binomial theorem, we have $\sum\limits_{k=0}^\infty {x \choose k} a^k b^{x - k} = (a + b)^x.$ Substituting $a = b = 1$, we have $f (x) = 2^x$, if we define $f : N \mapsto N$, that is, at the positive integer values of $x$.