Irreducibility of Laguerre Polynomial Ln(−1−n−r)(x)

We consider the algebraic properties of Generalized Laguerre Polynomials for negative integral values given by Ln(−1−n−r)(x)=∑j=0n(n−j+rn−j)xjj!. For different values of r, this family gives polynomials which are of great interest. Improving on the earlier results of Hajir and Sell, we prove that Ln(−1−n−r) is irreducible and compute its Galois group for r≤22. Also we prove that Ln(−1−n−r) is irreducible and its Galois group contains An whenever n>r1.63e1.00008r.