  A class of Stieltjes functions of finite type is introduced. These satisfy
Widder's conditions on the successive derivatives up to some finite order, and
are not necessarily smooth. We show that such functions have a unique integral
representation, along some generic kernel which is a truncated Laurent series
approximating the standard Stieltjes kernel. We then obtain a two-to-one
correspondence, via the logarithmic derivative, between these functions and a
subclass of hyperbolically monotone functions of finite type. This
correspondence generalizes a representation of HCM functions in terms of two
Stieltjes transforms earlier obtained by the first author.
