Algebraic Properties of a Family of Generalized Laguerre Polynomials

Hajir, Farshid

Canadian journal of mathematics, Tome 61 (2009) no. 3, pp. 583-603

Résumé

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 $L_{n}^{(-1-n-r)}(x)=\Sigma _{j=0}^{n}\left( _{n-j}^{n-j+r} \right){{x}^{j}}/j!$ is a $\mathbb{Q}$ -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 fromrecent 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.

Détail

DOI : 10.4153/CJM-2009-031-6

Mots-clés : 11R09, 05E35

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