Transmutation operators associated with a Bessel type operator on the half line and certain of their applications

We consider a second-order singular differential operator L on the zalf line which generalizes the Bessel operator. We construct a pair transmutation operators between L and the second derivative operator d²/dx². Using these transmutation operators, we firstly establish a Paley-Wiener theorem for the Fourier transform associated to L, and secondly introduce a generalized convolution on [0, ∞[ tied to L. Furthermore, a generalization of the classical Sonine integral transform is built.