Infinitely Log-monotonic Combinatorial Sequences

2013/04/18 by Chen, William Y. C., Guo, Jeremy J. F., Wang, Larry X. W.
#05A20 #11B68 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1304.5160

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≥ 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≥ 0} is log-concave. Furthermore, we prove that if a sequence \a_n\_{n≥ k} is ratio log-concave, then the sequence \√[n]a_n\_{n≥ 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 \√[n]D_n\_{n≥ 1}.

Related

• Higher order log-monotonicity of combinatorial sequences
• Monotonicity and log-behavior of some functions related to the Euler Gamma function
• Some Ratio Monotonic Properties of a New Kind of Numbers introduced by Z.-W. Sun
• On multiple and infinite log-concavity
• On Ratio Monotonicity of a New Kind of Numbers Conjectured by Z.-W. Sun