Algebraic Properties of a Family of Generalized Laguerre Polynomials

http://dx.doi.org/10.4153/CJM-2009-031-6
Canad. J. Math. 61(2009), 583-603

  Published:2009-06-01
 Printed: Jun 2009

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers $r,n\geq 0$, we conjecture that $L_n^{(-1-n-r)}(x) = \sum_{j=0}^n \binom{n-j+r}{n-j}x^j/j!$ is a $\Q$-irreducible polynomial whose Galois group contains the alternating group on $n$ letters. That this is so for $r=n$ was conjectured in the 1950's by Grosswald and proven recently by Filaseta and Trifonov. It follows from recent work of Hajir and Wong that the conjecture is true when $r$ is large with respect to $n\geq 5$. Here we verify it in three situations: i) when $n$ is large with respect to $r$, ii) when $r \leq 8$, and iii) when $n\leq 4$. The main tool is the theory of $p$-adic Newton Polygons.