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.

中文翻译：

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关于格罗斯瓦尔德猜想的推广

摘要 我们概括了 Grosswald 的一个猜想，现在是由于 Filaseta 和 Trifonov 的定理，指出由 yn ( x ) 表示的贝塞尔多项式在每个 n 的有理数上具有关联的伽罗瓦群 S n。我们考虑广义贝塞尔多项式 yn , β ( x )，其中包含有趣的多项式族，其判别式为非零有理平方。我们表明，如果 β ≥ 0 且 n 足够大，则与 yn 、β ( x ) 相关联的伽罗瓦群总是包含 A n 。对于 β 0 ，伽罗瓦群几乎总是包含 A n 。进一步证明，对于β - 2 ，在abc 猜想的假设下，yn 的伽罗瓦群，β ( x ) 包含所有足够大的n 的A n 。使用这些结果，Filaseta、Finch 和 Leidy 以及第一作者关于 yn , β ( x ) 的判别式的早期工作，