Is there closed form of integral of gamma function with specific range

Question

I like to know is there closed form of below integral

$$\Gamma[20,a]=\int^{\infty}_{a}t^{19}e^{-t}dt $$

I can find closed form when integral range is 0 from inf.

But it is not easy to search what i like to know.

Thank you!

Answer 1

It is enough to use integration by parts or the following fact: if $p(x)$ is a polynomial, $$ \int p(x)e^{-x}\,dx = C-\left(p(x)+p'(x)+p''(x)+\ldots\right)e^{-x}. $$ So we have:

$$ \int_{a}^{+\infty}t^{19}e^{-t}\,dt = e^{-a}\sum_{k=0}^{19}\binom{19}{k}k! \,a^{19-k}.$$