A characterization of completely alternating functions

In this article, we characterize completely alternating functions on an abelian semigroup $S$ in terms of completely monotone functions on the product semigroup $S\times {\mathbb{Z}}_{+}$. We also discuss completely alternating sequences induced by a class of rational functions and obtain a set of sufficient conditions (in terms of it’s zeros and poles) to determine them. As an application, we show a complete characterization of several classes of completely monotone functions on ${\mathbb{Z}}_{+}^2$ induced by rational functions in two variables. We also derive a set of necessary conditions for the complete monotonicity of the sequence $\{{\prod }_{i=1}^k\frac{(n+a_i)}{(n+b_i)}{\}}_{n\in {\mathbb{Z}}_{+}},a_i,b_i\in (0,\mathrm{\infty })$