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

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)xjj! 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+1.2762logr).