pdf for non-central gamma distribution I have a given gamma distribution as:
$f(x;k,\theta) = \frac{1}{\Gamma(k)\theta^{k}}x^{k-1}e^{\frac{-x}{\theta}}$ and a non-centrality parameter $\delta$.
Now, I need to find the pdf of this non-central gamma distribution $f(x;k,\theta,\delta)$?
I have found an expression of this in a paper by Oliveira and Ferreira. However, the pdf expression is in terms of shape parameter and non-centrality parameter only, which is given as
$f(x;k,\delta) = \displaystyle\sum_{i=0}^{\infty}e^{\frac{-\delta}{2}}\left(\frac{\delta}{2}\right)^i \left[ \frac{1}{\Gamma(k+i)}e^{-x}x^{k+i-1}\right]$.
Is there an expression for pdf that incorporates x,$\theta$,k, and $\delta$? Or, any approximations to make the non-central distribution to the central distributions?

As far as my monte-carlo simulation and closed form expression match, the non-central gamma could be well approximated by Amoroso distribution i.e.,
$f(x;k,\theta,\delta) = \frac{1}{\theta^k\Gamma(\theta)}(x-\delta)^{k-1}e^{\left(\frac{-(x-\delta)}{\theta} \right)}$
where $\delta$ is the location parameter.
