The product ϕ_{λ}^{(α,β)}(t_{1})ϕ_{λ}^{(α,β)}(t_{2}) of two Jacobi functions is expressed as an integral in terms of ϕ_{λ}^{(α,β)}(t_{3}) with explicit non-negative kernel, when α≧β≧−1/2. The resulting convolution structure for Jacobi function expansions is studied. For special values of α and β the results are known from the theory of symmetric spaces.

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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.