LAUSR

Abstract We generalize a conjecture of Grosswald, now a theorem due to Filaseta and Trifonov, stating that the Bessel polynomials, denoted by y n ( x ) , have the associated Galois group S n over the rationals for each n. We consider generalized Bessel polynomials y n , β ( x ) which contain interesting families of polynomials whose discriminants are nonzero rational squares. We show that the Galois group associated with y n , β ( x ) always contains A n if β ≥ 0 and n sufficiently large. For β 0 the Galois group almost always contains A n . It is further shown that for β − 2 , under the hypothesis of the abc conjecture, the Galois group of y n , β ( x ) contains A n for all sufficiently large n. Using these results, an earlier work of Filaseta, Finch and Leidy and the first author concerning the discriminants of y n , β ( x ) , we are able to explicitly describe the instances where the Galois group associated with y n , β ( x ) is A n for all sufficiently large n depending on β.