Families of Laguerre polynomials with alternating group as Galois group

Abstract

For an arbitrary real number α and a positive integer n, the Generalized Laguerre Polynomials (GLP) is a family of polynomials defined byLn(α)(x)=(−1)n∑j=0n(n+α)(n−1+α)⋯(j+1+α)(n−j)!j!(−x)j. Following the work of Banerjee, Filaseta, Finch and Leidy [2] which described the set A0∪A∞ of integer pairs (n,α) for which the discriminant of Ln(α)(x) is a nonzero square, where A0 is finite and A∞ is explicitly given infinite set, it was conjectured by Banerjee in [1] that for α≠−1, the only pair (n,α)∈A∞ for which the associated Galois group of Ln(α)(x) is not An is (4,23). In this paper, we verify this conjecture for (α,n)∈A∞ with α∈{−2n,−2n−2,−2n−4}.In fact, we prove more general results concerning the irreducibility and Galois groups of the Generalized Laguerre polynomials Ln(α)(x) for n∈N and integers α such that α∈[−2n−4,−2n]. The case α=−2n−1 corresponds to the Bessel polynomials which have been studied earlier by Filaseta and Trifonov [7] and Grosswald [9]. Our ideas give a simpler proof of their results.