On the $\varkappa\,$th root of a Stieltjes moment sequence

TL;DR

This paper investigates conditions under which the $ u$th root of a Stieltjes moment sequence is also a Stieltjes moment sequence, establishing formulas relating their measures' supports and characterizing possible support sizes.

Contribution

It provides a new formula linking the supports of measures for a Stieltjes moment sequence and its root, and characterizes support size pairs for such sequences.

Findings

Established a formula connecting supports of measures for original and root sequences.

Analyzed the holes in the supports of these measures.

Characterized all pairs of support sizes for which such sequences exist.

Abstract

Stieltjes moment sequences $\{a_n\}_{n=0}^\infty$ whose $\varkappa\,$th roots $\{\sqrtk{a_n}\}_{n=0}^\infty$ are Stieltjes moment sequences are studied ($\varkappa$ is a fixed integer greater than or equal to 2). A formula connecting the closed supports of representing measures of $\{a_n\}_{n=0}^\infty$ and $\{\sqrtk{a_n}\}_{n=0}^\infty$ is established. The relationship between the holes of the supports of these measures is investigated. The set of all pairs $(M,N)$ of positive integers for which there exists a Stieltjes moment sequence whose square root is a Stieltjes moment sequence and both of them have representing measures supported on subsets of $(0,\infty)$ of cardinality $M$ and $N$, respectively, is described.