Generalized Stieltjes functions and their exact order

Abstract

The paper surveys the basic properties of generalized Stieltjes functions including some new ones. We introduce the notion of the exact Stieltjes order and give a criterion of exactness along with simple sufficient conditions and some prototypical examples. The paper includes an appendix, where we define the left sided Riemann-Liouville and the right sided Kober-Erdlyi fractional integrals of measures supported on half axis and give inversion formulas for them.