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i need to find the PDf of the summation of limited number of independent, iid, non-central chi-square random variables. I found out that there is a PDF formula for almost all the sums of random variables, except the non-central chi-square $RV_s$, so anyone knows what it may be?

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The non-central chi-squared distribution has characteristic function $$\varphi_{k,\,\lambda}\left( t\right):=\left( 1-2it\right)^{-\tfrac{k}{2}}\exp\frac{i\lambda t}{1-2it}.$$ Summing $n$ iids with this distribution obtains the characteristic function $\varphi_{k,\,\lambda}^n=\varphi_{kn,\,\lambda n}$. The pdf is therefore $$\dfrac{1}{2}e^{-\tfrac{x+n\lambda}{2}}\left(\dfrac{x}{n\lambda}\right)^{\tfrac{kn}{4}-\tfrac{1}{2}}I_{\tfrac{kn}{2}-1}\left( \sqrt{n\lambda x}\right).$$

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    This is a great result, can you please give me the source for this result to see the full derivation? Thank you.@J.G. – 2017-01-20
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    @user42138 The above Wikipedia page includes a proof of the pdf. The characteristic function is considered here: http://www.planetmathematics.com/CharNonChi.pdf – 2017-01-20
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    I have a question about $k$, the degree of freedom, how can we specify its value? can we put it ($k=2$) for example? – 2017-01-20
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    @user42138 In theory $k$ can have any positive value, although we're usually interested in $k\in\mathbb{N}$. – 2017-01-20
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    i see, so it is not wrong if i put it 2, right? i have to do this because it will be inside an integration with other terms, so it will simplify the integral if it has a value of 2, otherwise, the integral will be so complicated.@J.G. – 2017-01-20
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    @user42138 Your use case will probably either dictate $k$ or require it to be general. – 2017-01-20
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    Hello, i'm sorry to ask after a long time, but today i wanted to do the same for another RV, so when i came back here i found that there is something i don't understand, i found the CF of the product of these RV's as you suggested, but how did you transform the resulted formula into the PDF of their sum? can you just tell me this piece of information? Thank you a lot. – 2017-03-08
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    @user42138 It's not generally easy, but for this problem the cf we needed to convert was of the came form as the original one, just with changed parameters. PDFs can be written in terms of cfs with an immersion formula, but it might not be elementary. I recommend creating a question about the new problem you're considering. – 2017-03-08
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    I see, however, according to this case, i need to know how did you decided that the pdf of sum of non central, chi-square $RV_s$ is also a non central chi-square with $nk$ and $n\lambda$? i found the CF of the product, it was: $(1-2it)^{-kn/2} exp(\frac{itn\lambda}{(1-2it)})$, just please show me how to continue from here. Thank you. – 2017-03-08
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    @user42138 I just stared at the cfs. As a simpler example, if you add independent Normal variables you can see from the cfs that you get a Normal distribution and you add the means and variances. It only works for certain distribution families, but the non-central chi squared family works. – 2017-03-08
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    i just saw the derivation of the pdf of the sum of normal $RV_s$, i read this after finding the CF of the multiplication on Wikipedia " no two distinct distributions can both have the same characteristic function, so the distribution of X + Y must be just this normal distribution." can this be used as a proof for my case too ( i mean the non central chi-square case.)? is this what you mean? – 2017-03-09
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    @user42138 That's the logic. PDFs are unique because $f\left( x\right)=\int_\mathbb{R}\frac{\varphi\left( t\right)e^{-itx}}{2\pi}dt$. For many problems it's difficult to compute this integral or to spot an $f$ that yields the desired cf from $\varphi\left( t\right)=\int_\mathbb{R} f\left( t\right)e^{itx}dx$. – 2017-03-09
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    alright, thank you so much. – 2017-03-09