Bounds on order of indeterminate moment sequences

TL;DR

This paper studies the order of entire functions associated with indeterminate Hamburger moment sequences, providing explicit bounds and conditions under which the order can be precisely determined.

Contribution

It introduces explicit upper bounds for the order based on the parameters of the canonical system and identifies conditions where these bounds are tight, extending previous results.

Findings

Explicit upper estimate for the order in terms of canonical system parameters

Conditions under which upper and lower bounds coincide, allowing exact computation of the order

Examples where the order differs from previous lower estimates due to regularity assumptions

Abstract

We investigate the order $\rho$ of the four entire functions in the Nevanlinna matrix of an indeterminate Hamburger moment sequence. We give an upper estimate for $\rho$ which is explicit in terms of the parameters of the canonical system associated with the moment sequence via its three-term recurrence. Under a weak regularity assumption this estimate coincides with a lower estimate, and hence $\rho$ becomes computable. Dropping the regularity assumption leads to examples where upper and lower bounds do not coincide and differ from the order. In particular we provide examples for which the order is different from its lower estimate due to M.S.Liv\v{s}ic.