A Fixed-Point Worpitzky Identity and a Positive Binomial Transform for Type $B$ Involutions

Let $\mathcal I_n^B$ be the involutions of the hyperoctahedral group $\mathfrak B_n$, and let $\des^B$ denote the descent number with respect to the natural Coxeter order. We derive the fixed-point-refined Worpitzky identity \[ \sum_{n\ge0}\frac{\mathcal F_n(p,t)\,z^n}{(1-t)^{n+1}} =\sum_{m\ge0} \fr