# On the Galois group of Generalised Laguerre polynomials II

## TL;DR

This paper investigates the algebraic structure, specifically the Galois group, of certain generalized Laguerre polynomials with parameters related to negative integers, contributing to the understanding of their symmetries and algebraic properties.

## Contribution

It provides new insights into the Galois groups of generalized Laguerre polynomials with specific parameter choices, extending previous algebraic studies of these polynomials.

## Key findings

- Determined the Galois group structure for specific parameter cases.
- Extended algebraic understanding of Laguerre polynomials.
- Connected polynomial properties to algebraic symmetry concepts.

## Abstract

For real number $\alpha,$ Generalised Laguerre Polynomials (GLP) is a family of polynomials defined by \begin{align*} L_n^{(\alpha)}(x)=(-1)^n\displaystyle\sum_{j=0}^{n}\binom{n+\alpha}{n-j}\frac{(-x)^j}{j!}. \end{align*}These orthogonal polynomials are extensively studied in Numerical Analysis and Mathematical Physics. In 1926, Schur initiated the study of algebraic properties of these polynomials. We consider the Galois group of Generalised Laguerre Polynomials $ L_n^{(\frac{1}{2}+u)}(x^2)$ when $u$ is a negative integer.