I would like to simplify the following expression,
$$\sum_{n=1}^{x}ne^{-a}\frac{a^{x-n}}{(x-n)!}$$
where $x$ is an integer and $a<1$.
Is it possible to lose the sum?
An approximation for the sum will be also helpful.
2026-03-30 15:17:15.1774883835
Simplifying $\sum_{n=1}^{x}ne^{-a}\frac{a^{x-n}}{(x-n)!}$, where $x$ is an integer and $a<1$
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It can be written using the Incomplete Gamma function as $$ {\frac {{{\rm e}^{-a}}{a}^{x}-\Gamma \left( x,a \right) \left( a-x \right) }{\Gamma \left( x \right) }} $$