I really have two questions: One is about computing a PDF and the second is about how to sum a series involving $K_v(x)$ that the PDF in question seems to contain.
I have come across the following problem related to my research:
Compute the PDF of $(X_1+X_2)\times N$ where $X_1$ and $X_2$ are Nakagami-$m$ R.V.'s and $N$ is a zero -mean Gaussian R.V. with variance $\sigma^2$,
$f_{X_i}(x) = \frac {2 m^mx^{2m-1}}{\Gamma(m)\Omega^m} \exp \left ( - \frac{mx^2}{\Omega} \right) ~~\text{for}~i=1,2 $ and $f_N(x)=\frac{1}{\sqrt{2 \pi}\sigma}\exp\left (-\frac{x^2}{2\sigma^2} \right ).$
A number of research articles discuss the probability distribution for $X_1+X_2$, for example here. The PDF is given by
$ f_{X_1+X_2}(x) = \frac {4\sqrt{\pi}\Gamma(m)m^{2m}x^{4m-1}}{\Gamma^2(m)\Gamma\left(2m+\frac12\right)2^{4m-1}\Omega^{2m}} \exp \left ( - \frac{mx^2}{\Omega} \right)\times{}_1F_1\left(2m;2m+\frac12;\frac{mr^2}{2\Omega}\right) $
All random variables are independent
I do know the general strategy for computing the PDF/CDF of the product of two independent R.V.'s its just that computation here becomes very tedious. I have obtained one expression but am not sure of its validity. It has an infinite series containing the modified Bessel function of the 2nd kind:
$ \sum_{n=0}^\infty\frac{(2m)_n m^n}{(2m+1/2)_n n!}\left[\left(\frac{m}{2\Omega}\right)^{3/2}\frac{|z|}{\sigma}\right]^n K_{2m+n-1/2}\left(\sqrt{\frac{2m}\Omega}\frac{|z|}\sigma\right) $
I stongly suspect that the above series can be simplified to the generalized Hypergeometric Function or some expression thereof. I searched a number of handbooks of special functions but none have quite the same expression. I would greatly appreciate any pointers or identities to attack the problem of summing this series. And to begin with, does the PDF look like this?