Infinitely Log-monotonic Combinatorial Sequences

TL;DR

This paper introduces the concept of infinitely log-monotonic sequences, establishes their connection with completely monotonic functions, and proves that several important combinatorial sequences possess this property, including Bernoulli, Catalan, and Domb numbers.

Contribution

It defines infinitely log-monotonic sequences, links them to completely monotonic functions, and proves that many classical combinatorial sequences are infinitely log-monotonic.

Findings

Bernoulli, Catalan, and central binomial coefficient sequences are infinitely log-monotonic.

Sequences like derangement, Motzkin, Fine, Delannoy, polyhex, and Domb are ratio log-concave.

Confirmed Sun's conjecture on the log-concavity of the Domb numbers' root sequence.

Abstract

We introduce the notion of infinitely log-monotonic sequences. By establishing a connection between completely monotonic functions and infinitely log-monotonic sequences, we show that the sequences of the Bernoulli numbers, the Catalan numbers and the central binomial coefficients are infinitely log-monotonic. In particular, if a sequence $\{a_n\}_{n\geq 0}$ is log-monotonic of order two, then it is ratio log-concave in the sense that the sequence $\{a_{n+1}/a_{n}\}_{n\geq 0}$ is log-concave. Furthermore, we prove that if a sequence $\{a_n\}_{n\geq k}$ is ratio log-concave, then the sequence $\{\sqrt[n]{a_n}\}_{n\geq k}$ is strictly log-concave subject to a certain initial condition. As consequences, we show that the sequences of the derangement numbers, the Motzkin numbers, the Fine numbers, the central Delannoy numbers, the numbers of tree-like polyhexes and the Domb numbers are ratio log-concave. For the case of the Domb numbers $D_n$, we confirm a conjecture of Sun on the log-concavity of the sequence $\{\sqrt[n]{D_n}\}_{n\geq 1}$.