Log-convex and Stieltjes moment sequences

TL;DR

This paper proves that Stieltjes moment sequences are infinitely log-convex, introduces q-Stieltjes moment sequences of polynomials, and shows many combinatorial sequences possess these properties, settling a conjecture and proposing new problems.

Contribution

It establishes the infinite log-convexity of Stieltjes moment sequences, introduces q-Stieltjes moment sequences, and provides criteria for transformations preserving these properties.

Findings

Many combinatorial sequences are Stieltjes moment sequences.

The infinite log-convexity of Schr"oder numbers is confirmed.

Criteria for transformations preserving Stieltjes moment sequences are provided.

Abstract

We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) P\'olya frequency sequences are infinitely log-concave. We introduce the concept of $q$-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences. We provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Many well-known combinatorial sequences are shown to be Stieltjes moment sequences in a unified approach and therefore infinitely log-convex, which in particular settles a conjecture of Chen and Xia about the infinite log-convexity of the Schr\"oder numbers. We also list some interesting problems and conjectures about the log-convexity and the Stieltjes moment property of the (generalized) Ap\'ery numbers.