# Computing the PDF of a product of the sum of 2 Nakagmi-m R.V.s with a Normal R.V

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?

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Just to demystify things a little bit, I obtained the the last expression through a straight forward application of $\int_0^\infty\frac1xf_{X_1+X_2}(x)f_{N_1}(\frac{z}{x})dx$, a well-known result. Bessel function enter the picture by replacing ${}_1F_1(\cdot)$ by its series representation and interchanging integration and summation (assuming its valid). The identity $\Gamma_b(\alpha)=2b^{\alpha/2}K_\alpha(2\sqrt{b})$ where $\Gamma_b(\alpha)=\int_0^\infty t^{\alpha-1} e^{-t-b/t} dt$ was used. –  Iconoclast Oct 16 '11 at 3:52