Let $(q;q)\_n=(1-q)(1-q^2)\cdots(1-q^n)$ with $(q;q)\_0:=1$. Define a $q$-exponential by 
$$e(z;q)=\sum\_{n\geq0}\frac{z^n}{(q;q)\_n}.$$
There is a notion of $q$-Eulerian polynomials, see [the reference](https://arxiv.org/pdf/1201.4941.pdf). I like to introduce **$q$-Eulerian polynomial of type B** via the generating function
$$\sum\_{n\geq1}B\_n(t,q)\frac{z^n}{(q;q)\_n}=\frac{(e(z;q)-e(tz;q))(e(tz;q)+te(z;q))}{e(2tz;q)-te(2z;q)}.$$
Now, expand $B\_n(t,q)$ as a polynomial 
$$B\_n(t,q)=\sum\_{k=0}^nB\_{n,k}(q)t^k$$
and call $B\_{n,k}(q)$ **$q$-Eulerian numbers type B**. Here are the first few terms:
\begin{align} B\_1(t,q)&=1+t, \\
B\_2(t,q)&=1+(2q+4)t+t^2, \\
B\_3(t,q)&=1+(7q^2+7q+9)t+(7q^2+7q+9)t^2+t^3.
\end{align}



> 
> **Claim.** if $a, b\in\Bbb{N}$ and $\alpha=a+b+1$, then the symmetric relation holds:
>  $$\binom{\alpha}a\_q+\sum\_k\binom{\alpha}k\_q2^{\alpha-k}B\_{k,b}(q)=
> \binom{\alpha}b\_q +\sum\_k\binom{\alpha}k\_q2^{\alpha-k}B\_{k,a}(q).$$
> 
> 
> 


**QUESTIONS:**


(a) I don't have a proof for my claim which seems very true though. Do you?


(b) Is there a combinatorial interpretation for these polynomials $B\_n(t,q)$ or the Eulerian numbers $B\_{n,k}(q)$? You might be inspired by [the reference](https://arxiv.org/pdf/1201.4941.pdf).



$q$-Eulerian type B enjoy symmetry
$\bf{Step~1}.$ $B\_{n,a}(q)=B\_{n,n-a}(q)$.


$\it{Proof}$. Write
$$
\sum\_{n\geq1}\dfrac{B\_n(t,q)}{t^{n/2}}\frac{z^n}{(q;q)\_n}=\frac{e(z/\sqrt{t};q)-e(z\sqrt{t};q)}{\dfrac{e(2z\sqrt{t};q)}{\sqrt{t}}-\sqrt{t}~e(2z/\sqrt{t};q)}\left(\dfrac{e(z\sqrt{t};q)}{\sqrt{t}}+\sqrt{t}~e(z/\sqrt{t};q)\right).
$$
The rhs is symmetric under $t\to 1/t$, so $\dfrac{B\_n(t,q)}{t^{n/2}}$ is also symmetric:
$$
\dfrac{B\_n(t,q)}{t^{n/2}}=t^{n/2}{B\_n(1/t,q)}.\tag{1}
$$
We know that $B\_n(t,q)=B\_{n,0}t^0+...$ starts with the power $t^0$. Due to $(1)$ this means that the highest power of $t$ in the polynomial $B\_n(t,q)$ is $n$, and $B\_{n,a}(q)=B\_{n,n-a}(q)$ as required.


$\bf{Step~2}.$ $\displaystyle{\binom{\alpha}a\_q+\sum\_k\binom{\alpha}k\_q2^{\alpha-k}B\_{k,b}(q)=
\binom{\alpha}b\_q +\sum\_k\binom{\alpha}k\_q2^{\alpha-k}B\_{k,a}(q)},~ a,b,\in \mathbb{N},~a+b+1=\alpha.$


$\it{Proof}$.
According to q-binomial theorem $e(z;q)=\sum\_{n\geq0}\frac{z^n}{(q;q)\_n}=\frac{1}{(z;q)\_\infty}$. Now we equate the coefficients of $z^n$ in both sides of 
\begin{align}
\left(\sum\_{l\geq0}\frac{(2tz)^l}{(q;q)\_l}-t\sum\_{l\geq0}\frac{(2z)^l}{(q;q)\_l}\right)\cdot\sum\_{k\geq1}B\_k(t,q)\frac{z^k}{(q;q)\_k}={(e(z;q)-e(tz;q))(e(tz;q)+te(z;q))}\\
=\left(\sum\_{l\geq0}\frac{z^l}{(q;q)\_l}-\sum\_{l\geq0}\frac{(tz)^l}{(q;q)\_l}\right)\left(\sum\_{l\geq0}\frac{(tz)^l}{(q;q)\_l}+t\sum\_{l\geq0}\frac{z^l}{(q;q)\_l}\right)
\end{align}
and obtain
$$
\sum \_{k=1}^n \binom{n}{k}\_q\frac{B\_k(t,q) \left(t^{n-k}-t\right) }{2^{k-n}}-\sum \_{k=1}^n \left(1-t^k\right) \left(t^{n-k}+t\right) \binom{n}{k}\_q=0.
$$
The coefficient of $t^{a+1}$ in the above formula is (note that for $a\in \mathbb{N}$ we have $a+1\neq1$ and for $b=n-a-1\in \mathbb{N}$ we have $a+1\neq n$, so the 'boundary' terms with $t$ and $t^n$ don't make any contribution in the formula below)
$$
-\sum \_{k=1}^n\binom{n}{k}\_q \frac{B\_{k,a}(q) }{2^{k-n}}+\sum \_{k=1}^n \binom{n}{k}\_q\frac{B\_{k,a+1+k-n}(q)}{2^{k-n}}+\binom{n}{a}\_q-\binom{n}{b}\_q=0.
$$
Finally, taking into account $B\_{k,a+1+k-n}(q)=B\_{k,k-b}(q)=B\_{k,b}(q)$ we have
$$
\binom{n}{b}\_q+\sum \_{k=1}^n\binom{n}{k}\_q \frac{B\_{k,a}(q) }{2^{k-n}}=\binom{n}{a}\_q+\sum \_{k=1}^n \binom{n}{k}\_q\frac{B\_{k,b}(q)}{2^{k-n}}.
$$


