The irreducibility of all but finitely many Bessel polynomials
Grosswald conjectured that the Bessel polynomials \(y_n(x)= \sum_{j= 0}^n x^j(n+ j)!/2^j(n- j)! j!\) are all irreducible over the rationals. The author proves that this is indeed the case for all but finitely many (possibly 0) nonnegative integers \(n\). In theory, the method used in this paper (which relies on results about Newton polygons of polynomials) would yield an effective bound on the number of reducible \(y_n(x)\). However, such a bound would be too large to be able to check Grosswald's conjecture for the remaining cases. More generally, the author shows that for any integers \(a_0, a_1,\dots, a_n\) with \(|a_0|= |a_n|= 1\) the polynomials \(\sum_{j= 0}^n a_j x^j(n+ j)!/2^j(n- j)! j!\) are irreducible for all but finitely many \(n\). He conjectures that this is even true for all \(n\).
scientific article; zbMATH DE number 811810