Why are solutions in Chapman-Enskog formalism expressed via Sonine polynomials? Before I posit my question, I would like to thank anyone willing to have a crack at it. 
I'm currently writing an assignment concerning the Chapman-Enskog method of expanding the probability density function and using this approach to solve the Boltzmann equation, which appears in kinetic theory of gases. I have studied much of the standard material, e.g. Chapman and Cowling: The Mathematical Theory of Non-uniform Gases. However, this book and many other sources lack a satisfying explanation regarding the use of Sonine polynomials in writing the solutions.
I would like to be able to explain why does the use of Sonine polynomials arise in the Chapman-Enskog formalism. I certainly was not able to figure out if this method has a basis in some underlying mathematical principle (like how it seems ''obvious'' to express solutions to the Schrödinger equation as a Fourier series). Can anyone offer an answer?
 A: After a year I suddenly realise that I can answer the question :)
To put it succinctly, Sonine polynomials $S_{l+\frac{1}{2}}^{n}(v^2)$ are eigenfunctions of the linearized collision operator $L(\phi)$, which appears in the linearized version of the Boltzmann equation:
\begin{equation}
\frac{\partial \phi}{\partial t} + v_{i}\frac{\partial \phi}{\partial r_i} = L(\phi)
\end{equation}
which we obviously obtain by insterting the Chapman Enskog series into the Boltzmann equation and truncating at the linear term ($f = f_{M}(1+\epsilon\phi)$, where $f_{M}$ is the Maxwell-Boltzmann distribution ($\epsilon$ is assumed to be of order unity - equal to one)).
This and much more is explained in greater detail in the book ''An introduction to the Theory of the Boltzmann equation'' by Stewart Harris (https://www.amazon.com/Introduction-Theory-Boltzmann-Equation-Physics/dp/0486438317), which I would recommend as a quality reference to anyone interested in the Boltzmann equation and kinetic theory in general.
