Index integral transformations of Titchmarsh type

In 1946 Titchmarsh [4] introduced the integral transformation g(τ)=∫o∞ReJiτ(x)f(x)dx, which depends on the index of a Bessel function, in connection with a continuous spectral Bessel function expansion in Sturm-Liouville boundary value problems. Here, we generalize this transformation by using the composition properties and a relationship with the Kontorovich-Lebedev and the Mellin-type transformations, and we give a variety of index transformations with a linear combination depending on a parameter of real and imaginary parts of a Bessel function. As it is shown the inversion formula consists of the integral over the index of a Lommel function.