  We study $\beta$-Jacobi diffusion processes on alcoves in $\mathbb R^N$,
depending on 3 parameters. Using elementary symmetric functions, we present
space-time-harmonic functions and martingales for these processes
$(X_t)_{t\ge0}$ which are independent from one parameter. This leads to a
formula for $\mathbb E(\prod_{i=1}^N (y-X_{t,i}))$ in terms of classical Jacobi
polynomials. For $t\to\infty$ this yields a corresponding formula for Jacobi
ensembles and thus Aomoto's Selberg integral.
