A variant of the Stieltjes transform on distributions of compact support

In this paper, by means of a generalized Lambert transform and the Möbius numbers, we obtain an inversion formula for a variant of the Stieltjes transform introduced by Goldberg [R.R. Goldberg, An inversion of the Stieltjes transform, Pacific J. Math. 8 (1958) 213–217.] and defined byG(x)=∫0∞f(t)tx2+t2dt,x>0on the space E′((0,∞))E′((0,∞)) of distributions of compact support. We also study the analyticity of G.