[Paper Review] Higher order log-monotonicity of combinatorial sequences

Bao-Xuan Zhu|arXiv (Cornell University)|Sep 24, 2013

Advanced Combinatorial Mathematics15 references3 citations

TL;DR

This paper introduces a novel method to analyze higher-order log-monotonicity in combinatorial sequences using a three-term recurrence relation. It proves that the ratio sequences of derangement, Motzkin, Fine, Franel, and Domb numbers are ratio log-convex, and establishes infinite log-monotonicity for derangement numbers and generalized binomial coefficients under specific conditions, extending prior results on Catalan and central binomial coefficients.

ABSTRACT

A sequence $\{z_n\}_{n\geq0}$ is called ratio log-convex in the sense that the ratio sequence $\{\frac{z_{n+1}}{z_n}\}_{n\geq0}$ is log-convex. Based on a three-term recurrence for sequences, we develop techniques for dealing with the ratio log-convexity of ratio sequences. As applications, we prove that the ratio sequences of numbers, including the derangement numbers, the Motzkin numbers, the Fine numbers, Franel numbers and the Domb numbers are ratio log-convex, respectively. Finally, we not only prove that the sequence of derangement numbers is asymptotically infinitely log-monotonic, but also show some infinite log-monotonicity of some numbers related to the Gamma function, in particular, implying two results of Chen {\it et al.} on the infinite log-monotonicity of the Catalan numbers and the central binomial coefficients.

Motivation & Objective

• To develop a systematic method for analyzing higher-order log-monotonicity in combinatorial sequences.
• To investigate the ratio log-convexity of ratio sequences for key combinatorial numbers such as derangement, Motzkin, Fine, Franel, and Domb numbers.
• To extend the infinite log-monotonicity results of Chen et al. from Catalan and central binomial coefficients to derangement numbers and generalized binomial coefficients.
• To provide a unified framework for verifying log-monotonicity of sequences defined by linear recurrences with variable coefficients.

Proposed method

• Develops a recurrence-based criterion for ratio log-convexity using a three-term linear recurrence: $ z_{n+1} = a_n z_n + b_n z_{n-1} $.
• Introduces a functional inequality condition involving $ f(x) = [(a_{n+1}a_n + b_{n+1})x + a_{n+1}b_n](x - a_{n-1})x^6 - b_{n-1}(a_n x + b_n)^4 $ to verify ratio log-convexity.
• Applies the criterion to prove that the ratio sequences of derangement, Motzkin, Fine, Franel, and Domb numbers are ratio log-convex.
• Uses the result of Chen et al. that if $ [ ext{log} f(x)]^{(n)} $ is completely monotonic, then the sequence $ f(n) $ is infinitely log-monotonic.
• Applies this criterion to generalized binomial coefficients $ C_i = \frac{(n_0 + ia)!}{(k_0 + ib)! (\overline{k_0} + i\overline{b})!} $ under conditions $ a \geq b + \overline{b} $ and $ -1 \leq k_0 - (n_0+1)b/a \leq 0 $.
• Employs integral representations of polygamma functions to analyze the $ n $-th derivative of the logarithm of the gamma function expression.

Experimental results

Research questions

• RQ1 Are the ratio sequences of derangement, Motzkin, Fine, Franel, and Domb numbers ratio log-convex?
• RQ2 Is the sequence of derangement numbers infinitely log-monotonic?
• RQ3 Under what conditions is the generalized binomial coefficient sequence $ \binom{n_0 + id}{k_0 + i\delta} $ infinitely log-monotonic?
• RQ4 Can the infinite log-monotonicity of Catalan and central binomial coefficients be extended to other combinatorial sequences?
• RQ5 What conditions ensure that the ratio sequence of a linearly recurrent sequence is itself log-convex?

Key findings

• The ratio sequence of the derangement numbers is ratio log-convex, and the sequence is asymptotically infinitely log-monotonic.
• The ratio sequences of Motzkin, Fine, Franel, and Domb numbers are all ratio log-convex.
• The sequence of derangement numbers is proven to be infinitely log-monotonic, extending prior results on Catalan and central binomial coefficients.
• The generalized binomial coefficient sequence $ C_i = \frac{(n_0 + ia)!}{(k_0 + ib)! (\overline{k_0} + i\overline{b})!} $ is infinitely log-monotonic when $ a \geq b + \overline{b} $ and $ -1 \leq k_0 - (n_0+1)b/a \leq 0 $.
• The Fuss-Catalan numbers $ C_p(n) = \frac{1}{(p-1)n+1} \binom{pn}{n} $ are infinitely log-monotonic for all integers $ p \geq 2 $.