On a generalization of a conjecture of Grosswald

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