相关论文: Log-behavior of two sequences related to the ellip…

The Catalan-Larcombe-French sequence $\{P_n\}_{n\geq 0}$ arises in a series expansion of the complete elliptic integral of the first kind. It has been proved that the sequence is log-balanced. In the paper, by exploring a criterion due to…

Let $\{P_n\}_{n\geq 0}$ denote the Catalan-Larcombe-French sequence, which naturally came up from the series expansion of the complete elliptic integral of the first kind. In this paper, we prove the strict log-concavity of the sequence…

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…

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…

In recent years, Sun has proposed numerous conjectures regarding the log-concavity of root sequences $\{\sqrt[n]{a_n}}_{n\geqslant 1}$. We establish criteria for the asymptotic log-concavity of $\{\sqrt[n]{a_n}}_{n\geqslant 1}$ and the…

We study log-concavity properties of real sequences $(a_n)_{n \ge 0}$ satisfying a $d$-th order linear recurrence whose coefficients are linear functions of $n$; the so-called P-recursive (or holonomic) sequences. Writing the recurrence in…

A sequence $\{ a_n \}_{n \ge 0}$ is said to be asymptotically $r$-log-convex if it is $r$-log-convex for $n$ sufficiently large. We present a criterion on the asymptotical $r$-log-convexity based on the asymptotic behavior of $a_n…

Let $\{P_n\}$ be the Catalan-Larcombe-French numbers given by $P_0=1,\ P_1=8$ and $n^2P_n=8(3n^2-3n+1)P_{n-1}-128(n-1)^2P_{n-2}$ $(n\ge 2)$, and let $S_n=P_n/2^n$. In this paper we deduce congruences for $S_{mp^r}\pmod{p^{r+2}}$,…

We consider the higher order Tur\'an inequality and higher order log-concavity for sequences $\{a_n\}_{n \ge 0}$ such that \[ \frac{a_{n-1}a_{n+1}}{a_n^2} = 1 + \sum_{i=1}^m \frac{r_i(\log n)}{n^{\alpha_i}} + o\left( \frac{1}{n^{\beta}}…

We prove the log-concavity of the Fennessey-Larcombe-French sequence based on its three-term recurrence relation, which was recently conjectured by Zhao. The key ingredient of our approach is a sufficient condition for log-concavity of a…

In recent years, the log-concavity of $\{\sqrt[n]{S_n}\}_{n\geq 1}$ have been received a lot of attention. Very recently, Sun posed the following conjecture in his new book: the sequences $\{\sqrt[n]{a_n}\}_{n\geq 2}$ and $\{…

We consider $\ell$-log-momotonic sequences and Laguerre inequality of order two for sequences $\{a_n\}_{n \ge 0}$ such that \[ \frac{a_{n-1}a_{n+1}}{a_n^2} = 1 + \sum_{i=1}^m \frac{r_i(\log n)}{n^{\alpha_i}} + o\left( \frac{1}{n^{\beta}}…

We prove a lemma, which we call the Order Ideal Lemma, that can be used to demonstrate a wide array of log-concavity and log-convexity results in a combinatorial manner using order ideals in distributive lattices. We use the Order Ideal…

Let $\overline{p}(n)$ denote the overpartition function. In this paper, we obtain an inequality for the sequence $\Delta^{2}\log \ \sqrt[n-1]{\overline{p}(n-1)/(n-1)^{\alpha}}$ which states that \begin{equation*} \log…

The large $n$ behaviour of the hypergeometric polynomial $$\FFF{-n}{\sfrac12}{\sfrac12}{\sfrac12-n}{\sfrac12-n}{-1}$$ is considered by using integral representations of this polynomial. This ${}_3F_2$ polynomial is associated with the…

We show that the sequence of moments of order less than 1 of averages of i.i.d. positive random variables is log-concave. For moments of order at least 1, we conjecture that the sequence is log-convex and show that this holds eventually for…

We show that the large Cartesian powers of any graph have log-concave valencies with respect to a ffxed vertex. We show that the series of valencies of distance regular graphs is log-concave, thus improving on a result of (Taylor,…

The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q=1, to the Catalan numbers: 1, 1, 2, 5, 14,..., a log-convex sequence. We use a combinatorial interpretation of…

Menon's proof of the preservation of log-concavity of sequences under convolution becomes simpler when adapted to 2-sided infinite sequences. Under assumption of log-concavity of two 2-sided infinite sequences, the existence of the…

Let $p(n)$ denote the partition function. Desalvo and Pak proved the log-concavity of $p(n)$ for $n>25$ and the inequality $\frac{p(n-1)}{p(n)}\left(1+\frac{1}{n}\right)>\frac{p(n)}{p(n+1)}$ for $n>1$. Let $r(n)=\sqrt[n]{p(n)/n}$ and…