On the generalized Stieltjes transform of distributions (Q1084301)

We obtain the generalized Stieltjes transform: \[ F(x)=\frac{\Gamma (A)}{\Gamma (B)}\frac{\Gamma (\beta +1)}{x}\int^{\infty}_{0}(\frac{t}{x})^{\beta}_ 2F_ 1(A,\beta +1;B;-\frac{t}{x})f(t)dt \] where \(A=\beta +\eta +1\), \(B=\alpha +A\), \(\beta\geq 0\) and \(\eta >0\), by the iteration of the generalized Laplace transform, in distributional sense. We discuss the above transform as a special case of convolution transform (in distributional sense) and prove an inversion formula for the above transform in distributional sense.