Some properties of angular integrals

We find new representations for Itzykson-Zuber like angular integrals for arbitrary beta, in particular for the orthogonal group O(n), the unitary group U(n) and the symplectic group Sp(2n). We rewrite the Haar measure integral, as a flat Lebesge measure integral, and we deduce some recursion formulae on n. The same methods give also Shatashvili's type moments. Finally, we prove that, in agreement with Brezin and Hikami's observation, the angular integrals are linear combinations of exponentials whose coefficients are polynomials in the reduced variables (x(i) - x(j))(y(i) - y(j)).