I'm studying special functions, especially Jacobi functions, related to the rank one groups ($U(1, n; \mathbb{F})$ where $\mathbb{F}$ is $\mathbb{C}$ or $\mathbb{H}$, the skew-field of quaternions), Via T. Koornwinder's paper Jacobi Functions and Analysis on Noncompact Semisimple Lie Groups. In page 43 of this paper he says:
$\int_{S(\mathbb{F}^n)} f(Re (y_n) + i Im (y_n)) dy = \int_0^1\int_0^{\pi} f(r e^{i\psi}) dm_{\alpha, \beta}(r, \psi)$
where $dm_{\alpha, \beta}(r, \psi) = \frac{2\Gamma(\alpha + 1)}{\Gamma(1/2)\Gamma(\alpha - \beta)\Gamma(p + 1/2)}(1 - r^2)^{\alpha - \beta - 1}(r \sin \psi)^{2\beta}rdrd\psi$.
However, I have no idea how to derive it. Where can I find this sort of formula? Or even better the theory that gives this formula?