Abstra t: In the early 1950's, Emil Grosswald began investigating the irredu ibility of the Bessel polynomials yn(x) and onje tured the irredu ibility of yn(x) over the rationals for all positive integers n. This paper resolves the onje ture by establishing the irredu ibility of a lass of polynomials that in ludes the Bessel polynomials.

In Stanley [8], the author introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. In [9], the same author gives a conjectured combinatorial interpretation for the coefficients of the polynomials. Here, we prove the conjecture for the terms of highest degree.

In Stanley [8] the author introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. In [9] the same author gives a conjectured combinatorial interpretation for the coefficients of the polynomials. Here, we prove the conjecture for the terms of highest degree.

In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality matrix functional is characterized by a Pearson-type equation with two matrix polynomials of degree not greater than 2 and 1. The proofs are given for a general...