Generalized Stieltjes transform of Banch space valued distributions (Q1081081)

An extension of a generalized Stieltjes transform \[ F(x)=\frac{\Gamma b\Gamma (\beta +1)}{\Gamma (\alpha +b)}\frac{1}{x}\int^{\infty}_{0}(t/x)^{\beta}F(b,\beta +1;b+\alpha;-t/x)f(t)dt; \] b\(=\beta +\eta +1\), \(\beta\geq 0\), \(\eta >0\) to Banach space valued distributions (generalized functions) is provided. An inversion formula for the above transform is also proved in the distributional (Banach space valued) sense.