Signed $(p,q)$-enumerator identities for type B permutations

Let $B_n$ be the hyperoctahedral group of signed permutations, and for $\sigma\in B_n$ let $\mathsf{fwex}(\sigma)$, $\mathsf{\neg}(\sigma)$, $\mathsf{nest}_B(\sigma)$, and $\mathsf{cro}_B(\sigma)$ denote respectively the flag weak excedance, number of negative entries, type B nesting number, and type B crossing number. Let $R_{n}(t,p,q)$ be the $(p,q)$-derivative polynomial defined in the surrounding text. For $n\geq 1$, the signed type B enumerator identities.

$$\sum_{\sigma\in B_n}(-1)^{\left\lfloor \frac{\mathsf{fwex}(\sigma)}{2}\right\rfloor}t^{\mathsf{\neg}(\sigma)}p^{\mathsf{nest}_B(\sigma)}q^{\mathsf{cro}_B(\sigma)}=\begin{cases}(-1)^{\frac n2}(t+1)R_{n-1}(t,p,q)&\text{if $n$ is odd},\\(-1)^{\frac{n-1}{2}}(t-1)R_{n-1}(t,p,q)&\text{if $n$ is even},\end{cases}$$

and

$$\sum_{\sigma\in B_n}(-1)^{\left\lceil \frac{\mathsf{fwex}(\sigma)}{2}\right\rceil}t^{\mathsf{\neg}(\sigma)}p^{\mathsf{nest}_B(\sigma)}q^{\mathsf{cro}_B(\sigma)}=\begin{cases}(-1)^{\frac n2}(t-1)R_{n-1}(t,p,q)&\text{if $n$ is even},\\(-1)^{\frac{n+1}{2}}(t+1)R_{n-1}(t,p,q)&\text{if $n$ is odd}.\end{cases}$$

These identities give signed refinements of the joint crossing–nesting enumerator for type B permutations in terms of the $(p,q)$-derivative polynomials. The parser provides no resolution evidence, so the status is recorded as open.