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Good Morning! I need to solve this integral to deal with gamma distributions with different shape parameters. It seems that the solution is given by a modification of the Kummer Confluent Hypergeometric Function ...

$\int_{0}^{z}(z-x)^{q-1}x^{p-1}e^{-kx}dx$

$q$, $p$ are integers and $z > 0$.

Thanks!!!

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    Have you tried change of variables against: http://dlmf.nist.gov/13.4.E1 ? – 2017-02-24

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As always, with me, you need to check this.
Let:$y=\frac{x}{z};dx=z\cdot dy$
Your problem is then:
$z^{q-1+p}{\displaystyle \int_{0}^{1}\left(1-y\right)^{q-1}}y^{p-1}e^{y\cdot\left(-k\cdot z\right)}dy$
Thus the answer is:
$\frac{z^{q+p-1}}{\Gamma\left(p\right)\Gamma\left(q\right)}\cdot\mathbf{M}\left(p,p+q,-k\cdot z\right)$
By: dlmf.nist.gov/13.4.E1