Families of Laguerre polynomials with alternating group as Galois group
Article Type
Research Article
Publication Title
Journal of Number Theory
Abstract
For an arbitrary real number α and a positive integer n, the Generalized Laguerre Polynomials (GLP) is a family of polynomials defined by [Formula presented] 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.
First Page
387
Last Page
429
DOI
10.1016/j.jnt.2022.04.001
Publication Date
12-1-2022
Recommended Citation
Jindal, Ankita and Laishram, Shanta, "Families of Laguerre polynomials with alternating group as Galois group" (2022). Journal Articles. 2870.
https://digitalcommons.isical.ac.in/journal-articles/2870
