Is there a way to get a clean presentation of the orthogonal polynomials w.r.t. the Maxwell distribution <https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution> ?
If you separate the problems to an even and odd problems, the even part of the orthogonal basis behaves nice enough: one simply takes the quotient of the odd Hermite polynomials by x (since the Maxwell distribution is given by $Cx^2\exp(-x^2/2)$, and Hermite polynomials are orthogonal w.r.t. $\exp(-x^2/2)$) ; as for the even part, I described a semi-solution in the comment below. However, I have no ide how the two parts interact.


The analog question for Laguerre polynomials (i.e. when one moves from the measure $\exp(-x)$ to the measure $C\_{\alpha}x^{\alpha}\exp(-x)$ yields a very satisfying result in the form of generalized Laguerre polynomials. Due to the relation between Laguerre polynomials and Hermite polynomials, I'd expect something similar here.



What are the orthogonal polynomials w.r.t. Maxwell distribution
These are the socalled *Maxwell polynomials* $M\_n^{(p)}(x)$, see page 75 and following of [Spectral Methods in Chemistry and Physics.](https://books.google.nl/books?id=mO4XBgAAQBAJ) The coefficients for $p=2$, which is the case you need, are given in Table 2.14 of that reference.


![](https://ilorentz.org/beenakker/MO/Maxwell_poly.png)


