Random variable $X$ is a sum of $N$ squared Gamma RVs as: $$X=\sum_{i=1}^{N}X_i^2$$ where $X_i\sim Gam(k,\theta_i)$, the shape parameter $k$ is not an integer and scale parameters $\theta_i, \forall i$ may be distinct values.
As I know there are several results for the PDF/CDF of a sum of gamma RVs (iid or non iid), but I could not find any literature related to this problem.
If anyone knows a solution for the PDF/CDF of $X$, please share with me.
The square of the Gamma is a generalized gamma distribution. The sum of generalized gamma distributions was studied in https://www.researchgate.net/publication/4197204_On_the_Distribution_of_the_Sum_of_Generalized_Gamma_Variates_and_Applications_to_Satellite_Digital_Communications/link/02e7e51d95c310d34c000000/download