Asymptotics on Two-dimensional Recursion Sequences

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 ϵ.     

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.     

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.