Combinatorial relations among restricted and half Eulerian polynomials of types $A$, $B$, and $D$

In this paper, we study relations among several types of Eulerian polynomials from a combinatorial viewpoint. We establish an identity between the restricted Eulerian polynomials of types $A$ and $B$. As an application, we present a bijective proof of a new identity involving the Eulerian polynomials of type $A$ and type $B$, solving a recent open problem proposed by Zhang. Additionally, we derive an identity between the half Eulerian polynomials of type $B$ and type $D$. Using this identity, we further obtain another relation about the Eulerian polynomials of type $A$ and type $B$, as well as a recursive formula connecting the restricted Eulerian polynomials of type $D$ and Eulerian polynomials of types $A$ and $B$.