Algebraic Properties of a Family of Generalized Laguerre Polynomials

Abstract

Abstract. We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r , n ≥ 0, we conjecture that 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 fromrecent work of Hajir andWong that the conjecture is true when r is large with respect to n ≥ 5. Here we verify it in three situations: (i) when n is large with respect to r , (ii) when r ≤ 8, and (iii) when n ≤ 4. The main tool is the theory of p -adic Newton Polygons.