I want to analyze or approximate a random variable that is a sum of two scaled independent non central $\chi$-squared random variables with the same degrees of freedom.
For example, $$X = X_1 + a X_2$$ where $X_1 \sim \chi^2(k,\lambda_1)$ and $X_2 \sim \chi^2(k,\lambda_2)$, $k$ is the degree of freedom, $\lambda_1, \lambda_2$ are noncentral parameters.
Simulations show that pdf of $X$ has the form of non central $\chi$-squared, or Gamma, ... I did try using characteristic function $M_X(t) = M_{X_1}(t)M_{X_2}(at)$ but it does not seem that the inverse transform of $M_X(t)$ has analytic form.
I wonder if there exists well-known results for this kind of problems.
For example,
$k=7, l_1=2,l_2=5, a=3$
$k=7, l_1=2,l_2=5, a=1$
$k=7, l_1=2,l_2=5, a=-1$
There is one case which is well-known.
Lemma. Let $Q_i \sim \chi^2_{k_i}(\lambda_i)$ for $i=1,\dots,n$, be independent. Then, $Q = \sum_{i=1}^n Q_i$ is a noncentral-$\chi^2_{k}(\lambda)$, where $k=\sum_{i=1}^n k_i$ and $\lambda = \sum_{i=1}^n \lambda_i$.
Proof. The Moment Generating Function (MGF) of $Q_i$ is given by $$ \mathcal{M}_{Q_i}(t) = (1 -2t)^{-k_i/2} \exp \Big\{ \frac{\lambda_i t}{1 - 2t} \Big\} $$ The MGF of $Q$, using independence, is given by $$ \mathcal{M}_Q(t) = \prod_{i=1}^n \mathcal{M}_{Q_i}(t) = \prod_{i=1}^n (1 -2t)^{-k_i/2} \exp \Big\{ \frac{\lambda_i t}{1 - 2t} \Big\} = (1 - 2t)^{- \frac{1}{2} \sum_{i=1}^n k_i} \exp \Big\{ \frac{t}{1 - 2t} \sum_{i=1}^n \lambda_i \Big\} $$ Thus, $$ \mathcal{M}_Q(t) = (1 -2t)^{-k/2} \exp \Big\{ \frac{\lambda t}{1 - 2t} \Big\} $$ and the thesis immediately follows. $\blacksquare$
Note that, if $\lambda \neq 0$, the PDF (probability density function) of $Q$ does not admit a closed-form, as it involves Bessel functions.
The general case of a linear combination of independent $\chi^2_{k_i}(\lambda_i)$ $$ Q = \sum_{i=1}^n a_i Q_i $$ results in a so-called generalized chi-squared distribution. The probability density function (PDF) of the generalized $\chi^2$ is also complicated.