# Hausdorff moment sequences induced by rational functions

## TL;DR

This paper investigates when sequences derived from rational functions are Hausdorff moments, establishing necessary and sufficient conditions, exploring connections with divided differences and convolutions, and applying results to polynomial sequences.

## Contribution

It provides new criteria for rational functions to generate Hausdorff moment sequences and answers a specific open question about subnormal modules.

## Key findings

- Derived necessary conditions for rational functions to produce Hausdorff moments.
- Established sufficient conditions based on zeros and poles of rational functions.
- Characterized polynomials up to degree 5 for which inverse sequences are Hausdorff moments.

## Abstract

We study the Hausdorff moment problem for a class of sequences, namely $(r(n))_{n\in\mathbb Z_+},$ where $r$ is a rational function in the complex plane. We obtain a necessary condition for such sequence to be a Hausdorff moment sequence. We found an interesting connection between Hausdorff moment problem for this class of sequences with finite divided differences and convolution of complex exponential functions. We provide a sufficient condition on the zeros and poles of a rational function $r$ so that $(r(n))_{n\in\mathbb Z_+}$ is a Hausdorff moment sequence. G. Misra asked whether the module tensor product of a subnormal module with the Hardy module over the polynomial ring is again a subnormal module or not. Using our necessary condition we answer the question of G. Misra in negative. Finally, we obtain a characterization of all real polynomials $p$ of degree up to $4$ and a certain class of real polynomials of degree $5$ for which the sequence $(1/p(n))_{n\in\mathbb Z_+}$ is a Hausdorff moment sequence.