Families of Laguerre polynomials with alternating group as Galois group

For an arbitrary real number alpha and a positive integer n , the Generalized Laguerre Polynomials (GLP) is a family of polynomials defined by L-n((alpha)) (x) = (-1)(n) sigma(n)(j=0) (n + alpha)(n - 1 + alpha) middotmiddotmiddot (j + 1 + alpha)/(n -j)!?! (-x)(j) Following the work of Banerjee, Filaseta, Finch and Leidy [2] which described the set A(0) boolean OR A(infinity) of integer pairs (n, alpha) for which the discriminant of L-n((alpha)) (x) is a nonzero square, where A(0) is finite and A(infinity) is explicitly given infinite set, it was conjectured by Banerjee in [1] that for alpha &NOTEQUexpressionL;-1, the only pair (n, alpha) is an element of A(infinity) for which the associated Galois group of L-n((alpha)) (x) is not A(n) is (4 , 23). In this paper, we verify this conjecture for (alpha, n) is an element of A(infinity) with alpha is an element of {-2n, -2n - 2 , -2n - 4}. In fact, we prove more general results concerning the irreducibility and Galois groups of the Generalized Laguerre polynomials L-n((alpha)) (x) for n is an element of N and integers alpha such that alpha is an element of [-2n - 4 , -2n]. The case alpha = -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. (C) 2022 Elsevier Inc. All rights reserved.