arXiv 1807.014122018-07math.CO

欧拉递归的渐近分布理论及其应用

An asymptotic distribution theory for Eulerian recurrences with applications

Hsien-Kuei Hwang, Hua-Huai Chern, Guan-Huei Duh

摘要：

本文研究了欧拉型线性递归式\[ P_n(v) = (α(v)n+γ(v))P_{n-1}(v) +β(v)(1-v)P_{n-1}'(v)\qquad(n\ge1), \]，给出了$P_0(v)$，其中$α(v), β(v)$和$γ(v)$在大多数情况下是低次多项式.我们的特点的各种极限法律的系数$P_n(v)$大$n$使用的矩量法和分析组合工具下不同的$α(v), β(v)$和$γ(v)$，并将我们的结果应用到200多个具体的例子时，$β(v)\ne0$和300多个时，$β(v)=0$，我们收集的文献和斯隆的OEIS数据库。我们得到的极限定律和收敛速度几乎都是新的，包括正态、半正态、Rayleigh、beta、Poisson、负二项、Mittag-Leffler、Bernoulli等，展示了这样一个简单的框架令人惊讶的丰富性和多样性，以及所使用的方法的力量。

英文摘要：

We study linear recurrences of Eulerian type of the form \[ P_n(v) = (α(v)n+γ(v))P_{n-1}(v) +β(v)(1-v)P_{n-1}'(v)\qquad(n\ge1), \] with $P_0(v)$ given, where $α(v), β(v)$ and $γ(v)$ are in most cases polynomials of low degrees. We characterize the various limit laws of the coefficients of $P_n(v)$ for large $n$ using the method of moments and analytic combinatorial tools under varying $α(v), β(v)$ and $γ(v)$, and apply our results to more than two hundred of concrete examples when $β(v)\ne0$ and more than three hundred when $β(v)=0$ that we gathered from the literature and from Sloane's OEIS database. The limit laws and the convergence rates we worked out are almost all new and include normal, half-normal, Rayleigh, beta, Poisson, negative binomial, Mittag-Leffler, Bernoulli, etc., showing the surprising richness and diversity of such a simple framework, as well as the power of the approaches used.

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摘要：

英文摘要：

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