Given the following p.d.f., which is the p.d.f. of the real and imaginary parts of a random variable that is the ratio between a complex Gaussian and a Chi-squared RVs:


\begin{equation\*}
 f\_U(u)=\exp\Big\{{-\frac{1}{4 u^2}}\Big\} \,\frac{\left(8 n u^2-1\right) I\_n\left(\frac{1}{4
 u^2}\right)+I\_{n+1}\left(\frac{1}{4 u^2}\right)}{4 |u|^3}, 
\end{equation\*}
where $n$ is a constant integer and $I\_n(z)$ is the modified Bessel function of the first kind.


I'd like to find its Cumulative distribution function (CDF).


Further information on this p.d.f. and how it was obtained can be found at: [Distribution of ratio between complex Gaussian and Chi-square R.V.s](https://mathoverflow.net/questions/290092/distribution-of-ratio-between-complex-gaussian-and-chi-square-r-v-s)


I've tried to solve with Mathematica:



```
ii = 1/Pi Integrate[w^(-1 - n) Exp[(-b*v)/(a*w)] 1/Sqrt[w - u^2], {w,u^2, \[Infinity]}, Assumptions -> (u > 0 && n > 2 && 1 > v > 0 && a > 0 && b > 0)]
jj = (b^n)/((n - 2)! a^n) Integrate[v^n (1 - v)^(n - 2) ii, {v, 0, 1}, Assumptions -> (u > 0 && n > 2 && a > 0 && b > 0)]
Integrate[jj, {u, -Infinity, t}, Assumptions -> (u > 0 && n > 2 && a > 0 && b > 0)]

```

but it gives out a solution with this Hypergeometric function. I wonder if there is a tricky or some other technique to find a simpler equation for the CDF.



CDF of a RV that is the ratio between a complex Gaussian and a Chi-squared RVs
The cumulative distribution function you are seeking is
$$F\_n(x)=2\int\_0^{x}\exp\left({-\frac{1}{4 u^2}}\right) \,\frac{\left(8 n u^2-1\right) I\_n\left(\frac{1}{4
 u^2}\right)+I\_{n+1}\left(\frac{1}{4 u^2}\right)}{4 u^3}\, du.$$
It has the general form
$$F\_n(x)=e^{-1/z}\bigl(A\_n(z)I\_0(1/z)+B\_n(z)I\_1(1/z)\bigr),$$
with $A\_n(z)$ a polynomial in $z=4x^2$ of degree $\max(0,n-3)$ and $B\_n(z)$ a polynomial in $z$ of degree $\max(0,n-2)$. For $n\in\{1,2,\ldots 10\}$ I find
$$\{A\_n\}=\left\{1,1,3,3-8 z,48 z^2-8 z+5,-384 z^3+48 z^2-32 z+5,3840 z^4-384 z^3+336 z^2-32 z+7,-46080 z^5+3840 z^4-4224 z^3+336 z^2-80 z+7,645120 z^6-46080 z^5+61440 z^4-4224 z^3+1296 z^2-80 z+9,-10321920 z^7+645120 z^6-1013760 z^5+61440 z^4-23424 z^3+1296 z^2-160 z+9\right\},$$
$$\{B\_n\}=\left\{0,2,2-4 z,16 z^2-4 z+4,-96 z^3+16 z^2-20 z+4,768 z^4-96 z^3+160 z^2-20 z+6,-7680 z^5+768 z^4-1632 z^3+160 z^2-56 z+6,92160 z^6-7680 z^5+19968 z^4-1632 z^3+736 z^2-56 z+8,-1290240 z^7+92160 z^6-284160 z^5+19968 z^4-11232 z^3+736 z^2-120 z+8,20643840 z^8-1290240 z^7+4608000 z^6-284160 z^5+192768 z^4-11232 z^3+2336 z^2-120 z+10\right\}.$$
(I checked that $\lim\_{x\rightarrow\infty}F\_n(x)=1$, as it should be.)


Perhaps these are known sequences of polynomials, I don't know.


