Series Expansion of the Incomplete Gamma Function

Question

The incomplete gamma function (upper) is defined as $\Gamma(a,x)=\int_x^{\infty}t^{a-1}e^{-t}dt$. Is there a series expansion for small non-integer $a << 1$ and small $x$?

Answer 1

Thanks to this question, we know that

$$\Gamma(a,x)=\Gamma(a)-\sum_{n=0}^\infty\frac{(-1)^nx^{a+n}}{n!(a+n)}$$

Which follows from the Taylor expansion of $e^x$.