Catalan-like numbers and Hausdorff moment sequences

TL;DR

This paper demonstrates that many classical combinatorial sequences are Hausdorff moment sequences, provides explicit measures for them, and explores conditions under which their subsequences and linear combinations also have this property.

Contribution

It unifies the understanding of Catalan-like numbers as moment sequences, proves their unique measures, and characterizes subsequences that preserve the moment property.

Findings

Catalan, Motzkin, binomial, and Delannoy numbers are Hausdorff moment sequences.

Explicit support intervals for their representing measures are determined.

Conditions for subsequences and linear combinations to be moment sequences are established.

Abstract

In this paper we show that many well-known counting coefficients, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the central Delannoy numbers are Hausdorff moment sequences in a unified approach. In particular we answer a conjecture of Liang at al. which such numbers have unique representing measures. The smallest interval including the support of representing measure is explicitly found. Subsequences of Catalan-like numbers are also considered. We provide a necessary and sufficient condition for a pattern of subsequences that if sequences are the Stieltjes Catalan-like numbers, then their subsequences are Stieltjes Catalan-like numbers. Moreover, a representing measure of a linear combination of consecutive Catalan-like numbers is studied.