Newton series representation of completely monotone functions

TL;DR

This work establishes a Newton series representation for completely monotone functions on a real right-unbounded interval, showing that such functions admit a Newton series expansion at every point $a$ with uniform convergence on compacta whenever a higher-order derivative is completely monotone, i.e., $f^{(q)}$ is CM. The authors develop a Newton interpolation framework based on forward differences and divided differences, proving a Newtonian analogue of Bernstein's little theorem and clarifying when the expansion exists (including counterexamples like $e^x$). They apply these results to principal indefinite sums $\Sigma g$, proving analytic and Newton-series expansions for a broad class of $g$ (e.g., $g(x)=\ln x$ and $g(x)=1/x$), and recover or generalize Bohr–Mollerup-type theorems such as the log-gamma function and Stern's series. The findings connect CM properties with discrete analytic representations, offering a discrete counterpart to Taylor expansions and new avenues for analyzing special functions via Newton series.

Abstract

We prove that every completely monotone function defined on a right-unbounded open interval admits a Newton series expansion at every point of that interval. This result can be viewed as an analog of Bernstein's little theorem for absolutely monotone functions. As an application, we use it to study principal indefinite sums, which are constructed via a broad generalization of Bohr-Mollerup's theorem.

Theorems & Definitions (26)

• Theorem 2.1: Bernstein's Little Theorem
• proof
• Remark 2.2
• Theorem 3.1
• Proposition 3.2
• proof
• Definition 3.3
• Proposition 3.4
• Lemma 3.5
• proof