Hurwitz Stability of Generalized Turán Expressions of Polynomial Sequences

In this paper we prove the weak Hurwitz stability conjecture for the strong Turán expressions of Eulerian polynomials, as well as the corresponding conjecture for Narayana polynomials, which were conjectured by Liu and Yan. We generalize the notion of Turán expressions of polynomial sequences, and provide a criterion for the generalized stability of generalized Turán expressions. Based on our sufficient condition, we not only confirm Liu and Yan’s conjectures but also obtain the generalized stability of generalized Turán expressions of a class of recurrent polynomial sequences and generalized Narayana polynomials.