二维递归序列的渐近性

In this paper, we give precise formulas for the general two-dimensional recursion sequences by generating function method, and make use of the multivariate generating functions asymptotic estimation technique to compute their asymptotic values.     

关于二项式系数倒数和幂和的渐近性

利用发生函数方法得到了涉及二项式系数倒数和幂和的恒等式并且应用发生函数的渐近计数方法给出了其渐近值.     

Stochastic aspects of the unitary dual group

In this note, we study the asymptotic properties of the ?-distribution of traces of some matrices, with respect to the free Haar trace on the unitary dual group. The considered matrices are powers of the unitary matrix generating the Brown algebra. We proceed in two steps, first computing the free cumulants of any R-cyclic family, then characterizing the asymptotic ?-distributions of the traces of powers of the generating matrix, thanks to these free cumulants. In particular, we obtain that these traces are asymptotic ?-free circular variables.     

Generating Functions and the Solutions of Full History Recurrence Equations

We study a class of full history recurrence equations using generating functions. The generating functions are shown to satisfy a first order linear differential equation. Singularity analysis and saddle-point methods can then be employed to derive asymptotic approximation for the coefficients of the generating functions.     

Asymptotic formulas related to the $$M_2$$M2-rank of partitions without repeated odd parts

The Ramanujan Journal - We give asymptotic expansions for the moments of the $$M_2$$-rank generating function and for the $$M_2$$-rank generating function at roots of unity. For this we apply the...     

Asymptotic enumeration of some RNA secondary structures

In this paper, we derive recursions of some RNA secondary structures with certain properties under two new representations. Furthermore, by making use of methods of asymptotic analysis and generating functions we present asymptotic enumeration of these RNA secondary structures.     

Nonlinear dispersive wave problems

In this paper we consider a certain class of nonlinear dispersive wave problems having solutions in the form of slowly varying wavetrains. We develop a procedure generating successively formal asymptotic approximations of these wavetrains of increasing asymptotic accuracy. In order to obtain formal asymptotic approximations we apply the two variable construction technique as developed in [3] for a class of perturbed oscillations described by nonlinear ordinary differential equations containing a small nonnegative perturbation parameter ?.     

Asymptotic distributions for Random Median Quicksort

The first complete running time analysis of a stochastic divide and conquer algorithm was given for Quicksort, a sorting algorithm invented 1961 by Hoare. We analyse here the variant Random Median Quicksort. The analysis includes the expectation, the asymptotic distribution, the moments and exponential moments. The asymptotic distribution is characterized by a stochastic fixed point equation. The basic technic will be generating functions and the contraction method.     

Euler数，Bernoulli数及有序Bell数的渐近估计

利用特殊函数的发生函数为亚纯函数的特点,给出Euler数、Bernoulli数、有序Bell数等几个组合数的带有余项的渐进估计．     

Words with a generalized restricted growth property

Words where each new letter (natural number) can never be too large, compared to the ones that were seen already, are enumerated. The letters follow the geometric distribution. Also, the maximal letter in such words is studied. The asymptotic answers involve small periodic oscillations. The methods include a chain of techniques: exponential generating function, Poisson generating function, Mellin transform, depoissonization.     

On Estimating the Cumulant Generating Function of Linear Processes

We compare two estimates of the cumulant generating function of a stationary linear process. The first estimate is based on the empirical moment generating function. The second estimate uses the linear representation of the process and the empirical moment generating function of the innovations. Asymptotic expressions for the mean square errors are derived under short- and long-range dependence. For long-memory processes, the estimate based on the linear representation turns out to have a better rate of convergence. Thus, exploiting the linear structure of the process leads to an infinite gain in asymptotic efficiency.     

有关调和数的更多渐近展开公式

基于指数型完全Bell多项式,建立了一个一般调和数渐近展开式,并给出展开式中系数的相应递推关系.由生成函数方法进一步推导出这些系数的具体表达式.另外,我们建立了两个在对数项里只含有奇数或偶数次幂项的lacunary调和数渐近展开式,     

Combinatorial sums through Riordan arrays

The aim of this work is to show how Riordan arrays are able to generate and close combinatorial identities, by means of the method of coefficients (generating functions). We also show how the same approach can be used to deal with other combinatorial problems, for instance asymptotic approximation and combinatorial inversion. Finally, we propose a method for generating new combinatorial sums by extending the concept of Riordan arrays to bi-infinite matrices.     

Stochastic Systems with Averaging in the Scheme of Diffusion Approximation

We propose a system approach to the asymptotic analysis of stochastic systems in the scheme of series with averaging and diffusion approximation. Stochastic systems are defined by Markov processes with locally independent increments in a Euclidean space with random switchings that are described by jump Markov and semi-Markov processes. We use the asymptotic analysis of Markov and semi-Markov random evolutions. We construct the diffusion approximation using the asymptotic decomposition of generating operators and solutions of problems of singular perturbation for reducibly inverse operators. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 9, pp. 1235–1252, September, 2005.     

The q-Catalan Numbers: A Saddle Point Approach

Using the saddle point method, we obtain from the generating function of the q-Catalan numbers and Cauchy’s integral formula asymptotic results in central and non-central regions.     

Analytic combinatorics of chord and hyperchord diagrams with k crossings

Using methods from Analytic Combinatorics, we study the families of perfect matchings, partitions, chord diagrams, and hyperchord diagrams on a disk with a prescribed number of crossings. For each family, we express the generating function of the configurations with exactly k crossings as a rational function of the generating function of crossing-free configurations. Using these expressions, we study the singular behavior of these generating functions and derive asymptotic results on the counting sequences of the configurations with precisely k crossings. Limiting distributions and random generators are also studied.     

Complexity of functions from some classes of three-valued logic

The problem of the realization complexity for functions of the three-valued logic taking values from the set {0, 1} by formulas over incomplete generating systems is considered. Upper and lower asymptotic estimates for the corresponding Shannon functions are obtained.     

Joint asymptotic multinormality for a class of rank order statistics in multivariate paired comparisons

The joint asymptotic multinormality of certain linear signed-rank statistics introduced by Shane and Puri (1969) is established for the nonidentically distributed case; moreover, the usual restriction forbidding constant score generating functions is dropped. In addition, sufficient conditions more general than those of Shane and Puri are given for the convergence of certain dispersion matrices, and these conditions guarantee the asymptotic independence of the statistics under consideration.     

Distributions of the Numbers of Failures and Successes in a Waiting Time Problem

Abstract. In this article we consider infinite sequences of Bernoulli trials and study the exact and asymptotic distribution of the number of failures and the number of successes observed before the r-th appearance of a pair of successes separated by a pre-specified number of failures. Several formulae are provided for the probability mass function, probability generating function and moments of the distribution along with some asymptotic results and a Poisson limit theorem. A number of interesting applications in various areas of applied science are also discussed.     

Several improved asymptotic normality criteria and their applications to graph polynomials

As widely regarded, one of the most classical and remarkable tools to measure the asymptotic normality of combinatorial statistics is due to Harper's real-rooted method proposed in 1967. However, this classical theorem exists some obvious shortcomings, for example, it requests all the roots of the corresponding generating function, which is impossible in general.Aiming to overcome this shortcoming in some extent, in this paper we present an improved asymptotic normality criterion, along with several variant versions, which usually just ask for one coefficient of the generating function, without knowing any roots. In virtue of these new criteria, the asymptotic normality of some usual combinatorial statistics can be revealed and extended. Among which, we introduce the applications to matching numbers and Laplacian coefficients in detail. Some relevant conjectures, proposed by Godsil (Combinatorica, 1981) and Wang et al. (J. Math. Anal. Appl., 2017), are generalized and verified as corollaries.