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
Recommended Citation
Jindal, Ankita; Nair, Saranya G.; and Shorey, Tarlok N., "Extension of irreducibility results on generalized Laguerre polynomials L(-n-s-1)n)(x)" (2024). Journal Articles. 4778.
https://digitalcommons.isical.ac.in/journal-articles/4778