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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 $x$?

David Young
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    You want an expansion around $x=0$ assuming $a \in (0,1)$ ? Did you notice that $\frac{\partial}{\partial x} \Gamma(a,x) = -x^{a-1} e^{-x}$ ? – reuns Feb 27 '17 at 23:38

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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 $e^x$.

Simply Beautiful Art
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