相关论文: Log-behavior of two sequences related to the ellip…
By making use of the familiar Mathieu series and its generalizations, the authors derive a number of new integral representations and present a systematic study of probability density functions and probability distributions associated with…
We prove that the overpartition function is log-concave for all n>1. The proof is based on Sills Rademacher type series for the overpartition function and inspired by Desalvo and Pak's proof for the partition function.
In this paper, we obtain some new inequalities for functions whose second derivatives' absolute value is s-convex and log-convex. Also, we give some applications for numerical integration.
Many famous integer sequences including the Catalan numbers and the Motzkin numbers can be expressed in the form $ConstantTermOf\left[P(x)^nQ(x)\right]$ for Laurent polynomials $Q$, and symmetric Laurent trinomials $P$. In this paper we…
Inversion sequences are integer sequences $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i}<i$ for each $i$. The study of patterns in inversion sequences was initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck in the…
A class of log-trigonometric integrals are evaluated in terms of elliptic functions. From this, by using the elliptic integral singular values, one can obtain closed form evaluations of integrals such as \[…
We formulate conditions on a set of log-concave sequences, under which any linear combination of those sequences is log-concave, and further, of conditions under which linear combinations of log-concave sequences that have been transformed…
The classical Fibonacci sequence is known to exhibit many fascinating properties. In this paper, we explore the Fibonacci sequence and integer sequences generated by second order linear recurrence relations with positive integer…
In this paper, we are interested in the long-time behaviour of stochastic systems of n interacting vortices: the position in R2 of each vortex evolves according to a Brownian motion and a drift summing the influences of the other vortices…
Let $t_n = (-1)^{s_2(n)}$, where $s_2(n)$ is the sum of binary digits function. The sequence $(t_n)_{n\in \mathbb N}$ is the well-known Prouhet-Thue-Morse sequence. In this note we initiate the study of the sequence $(h_n)_{n\in \mathbb…
We study the linear map sending the numerator of the rational function representing the Hilbert series of a module to that of its r-th Veronese submodule. We show that the asymptotic behaviour as r tends to infinity depends on the…
An inversion sequence of length $n$ is an integer sequence $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i}<i$ for each $i$. Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck began the study of patterns in inversion sequences,…
We investigate log-concavity in the context of level Hilbert functions and pure $O$-sequences, two classes of numerical sequences introduced by Stanley in the late Seventies whose structural properties have since been the object of a…
Recently, Z. W. Sun introduced a new kind of numbers $S_n$ and also posed a conjecture on ratio monotonicity of combinatorial sequences related to $S_n$. In this paper, by investigating some arithmetic properties of $S_n$, we give an…
Let $\overline{p}(n)$ denote the overpartition function. In this paper, we study the asymptotic growth of finite difference of logarithm of $\sqrt[n]{\overline{p}(n)/n^{\alpha}}$ for $\alpha$ being a non-negative real number, namely…
We prove log-concavity for the function counting partitions without sequences. We use an exact formula for a mixed-mock modular form of weight zero, explicit estimates on modified Kloosterman sums and analytic techniques. Finally, we…
Let K be a number field containing the n-th roots of unity for some n > 2. We prove a uniform subconvexity result for a family of double Dirichlet series built out of central values of Hecke L-functions of n-th order characters of K. The…
Denote by $A(p, n, k)$ the number of commuting $p$-tuples of permutations on $[n]$ that have exactly $k$ distinct orbits. It was conjectured in~\cite{abdesselam2023log} that $A(p, n, k)$ is log-concave with respect to $k$ for every $p\geq…
We survey some of the mechanisms used to prove that naturally defined sequences in combinatorics are log-concave. Among these mechanisms are Alexandrov's inequality for mixed discriminants, the Alexandrov Fenchel inequality for mixed…
In this paper we present many results and conjectures on congruences involving two types of Ap\'ery-like sequences $\{G_n(x)\}$ and $\{V_n(x)\}$.