Convolution of Hankel transform and its application to an integral involving Bessel functions of first kind (Q1895285)

The authors in this paper propose a definition of a convolution of the Hankel transform \[ {\mathcal H}_\nu [f] (x)= \int_0^\infty y J_\nu (xy) f(y) dy, \qquad \text{Re} (\nu)> -1/2, \] thereby proving the convolution property \[ {\mathcal H}_\nu [h] (x)= x^{-\nu} {\mathcal H}_\nu [f] (x) {\mathcal H}_\nu [g] (x), \] where \[ \begin{multlined} h(x)= {{2^{1- 3\nu} x^{-\nu}} \over {\sqrt {\pi} \Gamma (\nu+ 1/2)}} \int_{u+v>x} \int_{|u-v|<x} [x^2- (u- v)^2 ]^{\nu- 1/2} \times \\ \times [(u+ v)^2- x^2 ]^{\nu- 1/2} (uv)^{1- \nu} f(u) g(v) du dv. \end{multlined} \] The convolution is applied to evaluate an integral containing products of Bessel functions of the first kind.