On generalized absolutely monotone functions

Sufficient conditions for generalized absolutely monotone functions to possess a Taylor-type expansion in terms of the corresponding Extended Tchebycheff systems were found by Karlin and Ziegler. The question of necessary conditions, however, was left open. In this paper we solve this question by finding necessary and sufficient conditions for the validity of the expansion. The structure of the cone of generalized absolutely monotone functions and its extreme rays are also discussed. The research of the second author was partially supported by U.S. Army Contract-DA-31-124-ARO-D-462 in the MRC, Madison, Wisconsin.