On Expected Face Numbers of Random Beta and Beta' Polytopes

The random beta polytope is defined as the convex hull of $n$ independent random points with the density proportional to $(1-\mathrm{\parallel }x{\mathrm{\parallel }}^2{)}^{\beta }$ on the $d$-dimensional unit ball, where $\beta >-1$ is a parameter. Similarly, the random beta’ polytope is defined as the convex hull of $n$ independent random points with the density proportional to $(1+\mathrm{\parallel }x{\mathrm{\parallel }}^2{)}^{-\beta }$ on $\mathbb R^d$,where$\beta>\frac d2$. In a previous work [Angles of random simplices and face numbers of random polytopes, Adv. Math., 380 (2021), 107612], we established exact and explicit formulae for the expected $f$-vectors of these random polytopes in terms of certain definite integrals. In the present paper, we use purely algebraic manipulations to derive several identities for these integrals which yield alternative formulae for the expected $f$-vectors. Similar algebraic manipulations apply to Stirling numbers and yield the following identity: $$\sum _{s=0}^k\left\{\genfrac{}{}{0px}{}{n-s}{d-s}\right\}(d-s)\left[\genfrac{}{}{0px}{}{d-s}{k-s}\right]=\sum _{s=0}^k(-1{)}^s\left\{\genfrac{}{}{0px}{}{n-s}{d}\right\}\left[\genfrac{}{}{0px}{}{d+1}{k-s}\right]=\sum _{s=0}^{d-k}(-1{)}^s\left\{\genfrac{}{}{0px}{}{n+1}{d-s}\right\}\left[\genfrac{}{}{0px}{}{d-s}{k}\right]\mathrm{.}$$