Recurrence Ranks and Moment Sequences

TL;DR

This paper introduces new concepts of moment rank and unitary rank for sequences, linking them to various mathematical properties and solving key moment problems involving finite-atomic measures.

Contribution

It defines the moment and unitary ranks, characterizes them through multiple criteria, and solves the complex and integral finite-atomic moment problems.

Findings

Defined moment and unitary ranks for sequences.

Characterized these ranks via moments, recurrences, Hankel matrices, and algebraic properties.

Solved the complex and integer finite-atomic moment problems.

Abstract

We introduce the "moment rank" and "unitary rank" of numerical sequences, close relatives of linear-recursive order. We show that both parameters can be characterized by a broad set of criteria involving moments of measures, types of recurrence relations, Hankel matrix factorizations, Waring rank, analytic properties of generating functions, and algebraic properties of polynomial ideals. In the process, we solve the "complex finite-atomic" and "integral finite-atomic" moment problems: which sequences arise as the moments of a finite-atomic complex-/integer-valued measures on $\mathbb{C}$?