  In recent, H. Sun defined a new kind of refined Eulerian polynomials, namely,
\begin{eqnarray*} A_n(p,q)=\sum_{\pi\in \mathfrak{S}_n}p^{{\rm
odes}(\pi)}q^{{\rm edes}(\pi)} \end{eqnarray*} for $n\geq 1$, where
${odes}(\pi)$ and ${edes}(\pi)$ enumerate the number of descents of permutation
$\pi$ in odd and even positions, respectively. In this paper, we build an
exponential generating function for $A_{n}(p,q)$ and establish an explicit
formula for $A_{n}(p,q)$ in terms of Eulerian polynomials $A_{n}(q)$ and
$C(q)$, the generating function for Catalan numbers. In certain special case,
we set up a connection between $A_{n}(p,q)$ and $A_{n}(p,0)$ or $A_{n}(0,q)$,
and express the coefficients of $A_{n}(0,q)$ by Eulerian numbers. Specially,
this connection creates a new relation between Euler numbers and Eulerian
numbers.
