Algebraic properties of a family of Generalized Laguerre Polynomials

Details Algebraic properties of a family of Generalized Laguerre Polynomials Algebraic properties of a family of Generalized Laguerre Polynomials

2004-06-15

arxiv.org/abs/math/0406307v1

Mathematics

Number Theory

19 pages

Scientific paper

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 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 $n\geq 5$. 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.

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