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;R/ by the Hecke triangle groups. After fixing a normalization of the Haar measure on SL.2;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 R2 which arise as orbits of Hecke triangle groups. This generalizes work of W. Schmidt on the variance of the Siegel transform over SL.2; R/= SL.2; Z/.