I'm stuck with this convolution integral... \begin{equation} f_{Z}(z)=\int^{\infty}_{0}f_{1}(x)f_{2}(z-x)dx = \mbox{ } ??? \end{equation} which represents the pdf of the sum $Z = X_1 + X_2$ of two random variables $X_1$ and $X_2$. Each $X_1$ and $X_2$ belongs to the same family (each of them is proportional to a non-central $\chi^2$ variable): \begin{equation} \begin{aligned} f_{1}(x) &= \alpha_1 e^{-\beta_1 x} (\gamma_1 x)^{\frac{1}{2} \nu} I_{\nu}(\delta_1 \sqrt{x}) \\ f_{2}(x) &= \alpha_2 e^{-\beta_2 x} (\gamma_2 x)^{\frac{1}{2} \mu} I_{\mu}(\delta_2 \sqrt{x}) \end{aligned} \end{equation} where $I_{\nu}(\cdot)$ and $I_{\mu}(\cdot)$ are modified Bessel functions of the first kind of orders $\nu \geq 0$ and $\mu \geq 0$, respectively. Moreover, coefficients are all nonnegative: $a_i \geq 0, \beta_i \geq 0, \gamma_i \geq 0, \delta_i \geq 0$ and the
My questions:
1) can anybody provide a closed-form expression for the integral? Off course I know I can easily use Fourier, but I'm really wondering whether such pdf can be explicitly write down.
2) Moreover, is it possible to write $f_{Z}(z)$ as $f_1$ and $f_2$, i.e. do (?) $\left\{\alpha_z, \beta_z, \gamma_z, \delta_z, q \right\}$ exist such that \begin{equation} f_{Z}(z) = \alpha_z e^{-\beta_z z} (\gamma_z z)^{\frac{1}{2} q} I_{q}(\delta_z \sqrt{z}) \end{equation}
3) I'm separately interested in the case: $\nu = \mu = 0$.
Thanks for your attention.