Irreducibility and Galois Groups of Generalized Laguerre Polynomials $L_n^{(-1-n-r)}(x)$

Abstract

We study the algebraic properties of Generalized Laguerre polynomials for negative integral values of a given parameter which is $L_n^{(-1-n-r)}(x)=\sum _{j=0}^n\left(\genfrac{}{}{0ex}{}{n-j+r}{n-j}\right)\frac{x^j}{j!}$ for integers $r\ge 0,n\ge 1$. For different values of parameter $r$, this family provides polynomials which are of great interest. Hajir conjectured that for integers $r\ge 0$ and $n\ge 1$, $L_n^{(-1-n-r)}(x)$ is an irreducible polynomial whose Galois group contains $A_n$, the alternating group on $n$ symbols. Extending earlier results of Schur, Hajir, Sell, Nair and Shorey, we confirm this conjecture for all $r\le 60$. We also prove that $L_n^{(-1-n-r)}(x)$ is an irreducible polynomial whose Galois group contains $A_n$ whenever $n>e^{r\left(1+\frac{1.2762}{\log r}\right)}$.