Irreducibility and Galois groups of generalized Laguerre polynomials Ln^{(−1−n−r)}(x)

Authors

Ankita Jindal, Indian Institute of Technology Delhi
Shanta Laishram, Indian Statistical Institute (Delhi Centre)
Ritumoni Sarma, Indian Institute of Technology Delhi

Article Type

Research Article

Publication Title

Journal of Number Theory

Abstract

We study the algebraic properties of Generalized Laguerre polynomials for negative integral values of a given parameter which is Ln(−1−n−r)(x)=∑j=0n(n−j+rn−j)[Formula presented] for integers r≥0, n≥1. For different values of parameter r, this family provides polynomials which are of great interest. Hajir conjectured that for integers r≥0 and n≥1, Ln(−1−n−r)(x) is an irreducible polynomial whose Galois group contains An, the alternating group on n symbols. Extending earlier results of Schur, Hajir, Sell, Nair and Shorey, we confirm this conjecture for all r≤60. We also prove that Ln(−1−n−r)(x) is an irreducible polynomial whose Galois group contains An whenever n>er(1+[Formula presented]).

First Page

388

Last Page

406

DOI

10.1016/j.jnt.2017.08.003

Publication Date

2-1-2018

Comments

All Open Access, Bronze, Green

Recommended Citation

Jindal, Ankita; Laishram, Shanta; and Sarma, Ritumoni, "Irreducibility and Galois groups of generalized Laguerre polynomials Ln^{(−1−n−r)}(x)" (2018). Journal Articles. 1501.
https://digitalcommons.isical.ac.in/journal-articles/1501