$q$-Eulerian polynomials and polynomials with only real zeros

Abstract

Let $f$ and $F$ be two polynomials satisfying $F(x)=u(x)f(x)+v(x)f'(x)$. We characterize the relation between the location and multiplicity of the real zeros of $f$ and $F$, which generalizes and unifies many known results, including the results of Brenti and Br\"and\'en about the $q$-Eulerian polynomials.