Real-rooted polynomials via generalized Bell umbra

Abstract

In this paper, by the generalized Bell umbra and Rolle's theorem, we give some results on the real rootedness of polynomials. Some applications on partition polynomials are considered. Our results are illustrated by some comprehensive examples.

Derangement polynomials with a complex variable

“…For more information on the umbral calculus see [1,2,4,5,10,11]. In the remainder of this section, for any polynomials f and g, with integer coefficients we denote by f (z) ≡ g(z) to mean f (z) ≡ g(z) (mod pZ p [z]) and for any numbers a and b by a ≡ b we mean a ≡ b (mod p).…”