Jacobi functions: The addition formula and the positivity of the dual convolution structure

We prove an addition formula for Jacobi functions (varphi _lambda ^{left( {alpha ,beta } right)} left( {alpha geqq beta geqq - tfrac{1}{2}} right)) analogous to the known addition formula for Jacobi polynomials. We exploit the positivity of the coefficients in the addition formula by giving the following application. We prove that the product of two Jacobi functions of the same argument has a nonnegative Fourier-Jacobi transform. This implies that the convolution structure associated to the inverse Fourier-Jacobi transform is positive.