Taylor series of the upper incomplete gamma function

Question

What is the taylor series of the upper incomplete gamma function? I need it to approximate a difficult integration.

Answer 1

The Taylor series of the non-normalised upper incomplete gamma function is $$\Gamma(a,z) = \int_{z}^{\infty}t^{a-1}e^{-t}\,dt\\ = \Gamma(a)-\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{a+k}}{k!(a+k)} =\Gamma(a)\left(1-z^ae^{-z}\sum_{k=0}^{\infty}\frac{z^k}{\Gamma(a+k+1)}\right),$$ see e.g. http://dlmf.nist.gov/8.7.E3. If you need the normalised version, you have $$Q(a, z) = \frac{1}{\Gamma(a)}\int_{z}^{\infty}t^{a-1}e^{-t}\,d t =\frac{\Gamma(a,z)}{\Gamma(a)}\cdot$$