Theoretical Statistics and Mathematics Unit, ISI Delhi

On the Galois groups of generalized Laguerre Polynomials

by Shanta Laishram

For a positive integer $n$ and a real number $\alpha$, the generalized Laguerre polynomials are defined by
\begin{align*}
L^{(\alpha)}_n(x)=\sum^n_{j=0}\frac{(n+\alpha)(n-1+\alpha)\cdots (j+1+\alpha)(-x)^j}{j!(n-j)!}.
\end{align*}
These orthogonal polynomials are solutions to \emph{Laguerre's Differential Equation} which arises in the
treatment of the harmonic oscillator in quantum mechanics. Schur studied these Laguerre polynomials for its interesting algebraic properties.
In this short article, it is shown that the Galois groups of Laguerre polynomials
$L^{(\alpha)}_n(x)$ is $S_n$ with
$\alpha\in \{\pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{4}, \pm \frac{3}{4}\}$ except when
$(\alpha, n)\in \{(\frac{1}{4}, 2), (-\frac{2}{3}, 11), (\frac{2}{3}, 7)\}$. The proof is based on ideas of $p-$adic Newton polygons.

isid/ms/2015/13 [fulltext]

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