I am stuck in getting the following integral that involves product of lower incomplete gamma function with exponential function.
$\int_0^{\infty}\gamma(a,b\cdot q) e^{-K\cdot q}\text{dq}$
Any help in this regard is highly appreciated.
Note that $a,b$ and $K$ are constants.
$$I=\int_0^\infty\gamma(ab,q)\exp(-Kq)dq$$
$$ u=\gamma(ab,q)=\int_q^\infty t^{ab-1}e^{-t}dt\Rightarrow u'=-q^{ab-1}e^{-q} \\v'=\exp(-Kq)\Rightarrow v=-\exp(-Kq)/K $$ This is just integration by parts and leaves us with: $$I=\left.uv\right|_{0}^\infty-\int_0^\infty u'v \,dq$$ and this integral will be: $$\frac1K\int_0^\infty q^{ab-1}e^{-(K+1)q}dq$$ and now this is just the full gamma function. Just make sure to carefully evaluate $uv$