Simplify $\displaystyle \sum_{k=0}^{n}\frac{p^k}{k!}$

Question

Note that this is not a infinite sum like $\displaystyle\sum_{k=0}^{\infty}\frac{p^k}{k!}=e^p$, but a finite one.

Or, simplify the infinite sum $\displaystyle\sum_{k=0}^{\infty}\frac{p^k}{(k+n)!}$ may also help.

Maybe it can be solved by $Z$-transformation?

Answer 1

MathWorld calls this the exponential sum function, and notes that it can be expressed in terms of the incomplete gamma function. Indeed, $$\sum_{k=0}^n \frac{p^k}{k!} = \frac{e^p \Gamma(n+1,p)}{\Gamma(n+1)},$$ taken from https://mathworld.wolfram.com/ExponentialSumFunction.html.