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Is there a smooth function $f:\mathbb{R}^2 \to \mathbb{R}$ such that $f(x,n)$, where $n\in\mathbb{N}$, is the truncated Taylor series of $e^x$, namely $1+ x + \frac{x^2}{2} + \dotsb + \frac{x^n}{n!}$, in a fashion similar to the gamma function?

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You probably mean not only smooth but analytic? –  Slaviks Mar 29 '12 at 5:30

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up vote 6 down vote accepted

Sure; we have the relationship

$$\sum_{k=0}^n \frac{x^k}{k!}=\frac{\Gamma(n+1,x)}{\Gamma(n+1)}\exp\,x=Q(n+1,x)\exp\,x$$

where $\Gamma(n)$ is the usual gamma function and $\Gamma(n,a)$ is the (upper) incomplete gamma function, and $Q(n,a)$ is the regularized (upper) incomplete gamma function; see that link as well as this link for more on its properties. The expression on the right definitely makes sense not only for integer values of $n$...

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