1

In the wikipedia article Noncentral chi distribution the raw moments are given by Laguerre polynomials $L_n^{(a)}(z)$ with $n=1/2$ and $n=3/2$ but a Laguerre polynomial is only defined for $n \in \mathbb{N}$. How to understand this or how to correct the wikipedia article?

Citation from the wikipedia article:

===Raw moments===

The first few raw moments are:

$\mu^{'}_1=\sqrt{\frac{\pi}{2}}L_{1/2}^{(k/2-1)}\left(\frac{-\lambda^2}{2}\right)$

$\mu^{'}_2=k+\lambda^2$

$\mu^{'}_3=3\sqrt{\frac{\pi}{2}}L_{3/2}^{(k/2-1)}\left(\frac{-\lambda^2}{2}\right)$

$\mu^{'}_4=(k+\lambda^2)^2+2(k+2\lambda^2)$

where $L_n^{(a)}(z)$ is the generalized Laguerre polynomial.

Community
  • 1
granular_bastard
  • 517

1 Answers1

1

You're right, these aren't polynomials, they're more general Laguerre functions. Even though the Wikipedia article incorrectly called them “generalized Laguerre polynomials”, the underlying link correctly linked to the article on Laguerre functions, which allows for arbitrary real $n$. I've fixed the section you quoted by replacing “generalized Laguerre polynomial” by “Laguerre function”.

joriki
  • 238,052
  • Thanks. Is there a handy solution like a polynomial especially for $n=1/2$ ? – granular_bastard Dec 27 '19 at 23:29
  • @granularbastard: Unfortunately the functions for $n=\frac12$ seem to a be a bit messy. You can get them in Wolfram|Alpha (and probably in Mathematica) using LaguerreL(1/2,alpha,x). For instance, Wolfram|Alpha returns closed forms for LaguerreL(1/2,1/2,x), LaguerreL(1/2,1,x) and LaguerreL(1/2,3/2,x), but they're not polynomials. – joriki Dec 27 '19 at 23:51