Algebraic properties of a family of Generalized Laguerre Polynomials

Abstract

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers $r,n\ge 0$, we conjecture that $L_n^{(-1-n-r)}(x)={\sum }_{j=0}^n\left(\genfrac{}{}{0ex}{}{n-j+r}{n-j}\right)x^j/j!$ 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 50'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 $n\ge 5$. Here we verify it in three situations: i) when $n$ is large with respect to $r$, ii) when $r\le 8$, and iii) when $n\le 4$. The main tool is the theory of $p$-adic Newton Polygons.

19 pages

References in corpus (1)

• On the Galois group of Generalized Laguerre Polynomials

Cited by in corpus (1)

• On the Galois group of Generalized Laguerre Polynomials