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Sometimes there are functions that are initially defined for only integer values of the argument, but can be extended to functions of real variable by some obvious way. An example of this is the factorial ##n!## which is extended to a gamma function by a convenient integral definition.

So, if I have an arbitrary function ##f(x)## for which I know its value for any ##x\in\mathbb{N}##, is it possible to uniquely extend it to ##x\in\mathbb{R}## so that it becomes a ##\mathcal{C}^\infty## function?

Possible examples could be ##f(n) = 1 + 2 + \dots + n##, or the "power tower" function ##f(z,n) = z^{z^{\cdot^z}}## with ##n## ##z##:s in it.

Edit: For the first example it is straightforward to prove by induction that ##f(n) = \frac{n(n+1)}{2}## and then plug any value of ##n## in it, but for the second one it is not as obvious.

So, if I have an arbitrary function ##f(x)## for which I know its value for any ##x\in\mathbb{N}##, is it possible to uniquely extend it to ##x\in\mathbb{R}## so that it becomes a ##\mathcal{C}^\infty## function?

Possible examples could be ##f(n) = 1 + 2 + \dots + n##, or the "power tower" function ##f(z,n) = z^{z^{\cdot^z}}## with ##n## ##z##:s in it.

Edit: For the first example it is straightforward to prove by induction that ##f(n) = \frac{n(n+1)}{2}## and then plug any value of ##n## in it, but for the second one it is not as obvious.

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