Expected hyperbolic volumes of random beta polytopes

Let $X_1,\dots ,X_n$ be independent random points in the closed unit ball of ${\mathbb{R}}^d$. Assume that each $X_i$ has a beta distribution with parameter ${\beta }_i\ge -1$: if ${\beta }_i>-1$, then $X_i$ has Lebesgue density proportional to $(1-\mathrm{\parallel }x{\mathrm{\parallel }}^2{)}^{{\beta }_i}$ on $\{\mathrm{\parallel }x\mathrm{\parallel }<1\}$, whereas the case ${\beta }_i=-1$ corresponds to the uniform distribution on the unit sphere $\{\mathrm{\parallel }x\mathrm{\parallel }=1\}$. Let $[X_1,\dots ,X_n]$ denote the convex hull of these points. Interpreting the unit ball as the Klein model of hyperbolic geometry, we derive closed-form formulas for the expected hyperbolic volume of the random hyperbolic polytope $[X_1,\dots ,X_n]$. As a special case, if $X_1,\dots ,X_n$ are independent and uniformly distributed on the unit sphere in ${\mathbb{R}}^3$, then for every $n\ge 4$, $$\mathbb{E}{\mathrm{Vol}}_3^{\mathrm{h}\mathrm{y}\mathrm{p}}\text{\hspace{0.17em}⁣}([X_1,\dots ,X_n])=\pi (\frac{n}{2}-\sum _{j=1}^{n-1}\frac{1}{j})\mathrm{.}$$