Bounds on Order of Indeterminate Moment Sequences

We investigate the order ρ of the four entire functions in the Nevanlinna matrix of an indeterminate Hamburger moment sequence. We give an upper estimate for ρ 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 ρ 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. Livšic.