The convolution structure for Jacobi function expansions

The product ϕ ${}_{λ}^{(α,β)}$ (t1)ϕ ${}_{λ}^{(α,β)}$ (t2) of two Jacobi functions is expressed as an integral in terms of ϕ ${}_{λ}^{(α,β)}$ (t3) 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.