A higher moment formula for the Siegel--Veech transform over quotients by Hecke triangle groups

We compute higher moments of the Siegel–Veech transform over quotients of $SL(2,\mathbb{R})$ by the Hecke triangle groups. After fixing a normalization of the Haar measure on $SL(2,\mathbb{R})$ we use geometric results and linear algebra to create explicit integration formulas which give information about densities of $k$-tuples of vectors in discrete subsets of ${\mathbb{R}}^2$ which arise as orbits of Hecke triangle groups. This generalizes work of W.~Schmidt on the variance of the Siegel transform over $SL(2,\mathbb{R})\mathrm{/}SL(2,\mathbb{Z})$.