The quantity $(\lambda - z)^\rho $ is expanded in Jacobi polynomials $P_n^{(\alpha ,\beta )} (z)$, where $\alpha $, $\beta $, and $\rho $ are unrelated. The known case $\alpha = \beta = - \rho - 1$ is then used in a short proof of the addition theorem for Gegenbauer polynomials. The only other ingredients of the proof are well-known generating relations for these polynomials.