On Powers of the Catalan Number Sequence

TL;DR

This paper explores the fundamental properties of the Catalan number sequence, proving its infinite divisibility as a Stieltjes moment sequence and examining related sequences and conjectures in moment theory.

Contribution

It establishes the Catalan sequence as an infinitely divisible Stieltjes moment sequence and investigates powers and related sequences, introducing new theoretical insights.

Findings

Catalan sequence is an infinitely divisible Stieltjes moment sequence.

Any positive real power of the sequence remains Stieltjes determinate.

Includes analysis of related sequences like binomial coefficients and Fuss-Catalan.

Abstract

The Catalan number sequence is one of the most famous number sequences in combinatorics and is well studied in the literature. In this paper we further investigate its fundamental properties related to the moment problem and prove for the first time that it is an infinitely divisible Stieltjes moment sequence in the sense of S.-G. Tyan. Besides, any positive real power of the sequence is still a Stieltjes determinate sequence. Some more cases including (a) the central binomial coefficient sequence (related to the Catalan sequence), (b) a double factorial number sequence and (c) the generalized Catalan (or Fuss-Catalan) sequence are also investigated. Finally, we pose two conjectures including the determinacy equivalence between powers of nonnegative random variables and powers of their moment sequences, which is supported by some existing results.