# On a certain $(-1)$-Eulerian polynomials of type $B$

Let $$(q)_n=(1-q)(1-q^2)\cdots(1-q^n)$$ with $$(q)_0:=1$$. Define a $$q$$-exponential by $$e_q(z)=\sum_{n\geq0}\frac{z^n}{(q)_n}.$$ There is a notion of $$q$$-Eulerian polynomials of type $$A$$, see the reference. I like to introduce $$q$$-Eulerian polynomial of type B via the generating function $$\sum_{n\geq1}B_n(t,q)\frac{z^n}{(q)_n} =\frac{(e_q(z)-e_q(tz))\cdot(e_q(tz)+te_q(z))}{e_q(2tz)-te_q(2z)}.$$ 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. The first few terms are: \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} Here is an earlier MO problem. This time, I'm interested in a specialized aspect of it. For instance, $$B_n(t,1)$$ become the ordinary Eulerian polynomials of type $$B$$ whose coefficients are listed at OEIS. On the other hand, the polynomials $$B_n(t,-1)$$ do not appear anywhere. You may look at the first few of these: $$B_1(t,-1)=1, B_2(t,-1)=1+t, B_3(t,-1)=1+2t+t^2$$, $$B_4(t,-1)=1+9t+9t^2+t^3 \qquad\text{and} \qquad B_5(t,-1)=1+12t+22t^2+12t^3+t^4.$$

QUESTION 1. Is there some interpretation of the coefficients in $$B_n(t,-1)$$?

QUESTION 2. Can you provide a proof for the unimodality of $$B_n(t,-1)$$?