Extension of irreducibility results on generalized Laguerre polynomials L(-n-s-1)n)(x)

Article Type

Research Article

Publication Title

Publicationes Mathematicae Debrecen

Abstract

Let n be a positive integer. We consider the irreducibility of generalized Laguerre polynomials of the form L(-n-s-1) n (x) = Σn j=0 (-1)n n + s - j n - j ! xj j! . For different values of s, this family gives polynomials which are of great interest. It was proved earlier that for 0 ≤ s ≤ 60, these polynomials are irreducible over Q, and their Galois groups are shown to be An or Sn. In this paper, we prove that L(-n-s-1) n (x) is irreducible for each s ≤ 92. Also, we prove that L(-n-s-1) n (x) has no linear factor for each 93 ≤ s ≤ 100. Furthermore, assuming the irreducibility of L(-n-s-1) n (x) for 93 ≤ s ≤ 100, we determine that the Galois group of L(-n-s-1) n (x) is either An or Sn for each 61 ≤ s ≤ 100.

First Page

473

Last Page

494

DOI

10.5486/PMD.2024.9906

Publication Date

1-1-2024

Share

COinS