A characterization of completely alternating functions

TL;DR

This paper characterizes completely alternating functions on abelian semigroups using completely monotone functions, explores their relation with rational functions, and provides conditions for monotonicity in multivariate cases.

Contribution

It offers a new characterization of completely alternating functions via completely monotone functions and establishes conditions for their monotonicity based on rational function properties.

Findings

Characterization of completely alternating functions on abelian semigroups.

Conditions for complete monotonicity of sequences induced by rational functions.

Complete characterization of certain classes of completely monotone functions on .

Abstract

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 $\{\prod_{i=1}^{k}\frac{(n+a_i)}{(n+b_i)}\}_{n \in \mathbb Z_+}, a_i, b_i \in (0,\infty)$