Central Limit Theorem for local linear statistics in classical compact groups and related combinatorial identities

组合恒等式的泛化与概率证明

Generalization and Probabilistic Proofs of Some Combinatorial Identities

本文通过概率方法推导出涉及伽马和贝塔函数的组合恒等式，扩展了二项式系数和特殊函数的已知恒等式，并通过研究两个伽马和贝塔随机变量差的矩得出新恒等式。

Combinatorial Identities via Two Urn Models

We prove some combinatorial identities using the Polya urn and the closely related Hoppe urn.

来自非均匀伊辛链的组合恒等式

Combinatorial identities from an inhomogeneous Ising chain

该研究通过非均匀Ising链与最近邻粒子系统的对应关系，证明了与整数分拆、Frobenius分拆相关的组合恒等式，得到特定Ising链配分函数的乘积形式，并发现粒子系统的长程可逆动力学。

Exponential Order Statistics, the Basel problem and Combinatorial Identities

We consider the k-th order statistic from unit exponential distribution and show that it can be represented as a sum of independent exponential random variables. Our proof is simple and different. It readily proves that the standardized exponential spacings also follow unit exponential distribution. An interesting probabilistic proof of the Basel problem is also given. Another advantage of our approach is that by computing the Laplace transform of the k-th order statistic in two different ways, we derive several interesting combinatorial identities. A probabilistic interpretation of these iden…

Beta矩、组合恒等式与随机游走的统一方法

A Unified Approach to Beta Moments, Combinatorial Identities, and Random Walks

本文提出统一概率方法，将任意维简单对称随机游走的返回概率与矩表示关联，给出涉及Beta和Gamma函数的组合恒等式的概率证明，并推导新恒等式。

Random Walk Models for Nontrivial Identities of Bernoulli and Euler Polynomials

We consider the $1$-dimensional reflected Brownian motion and $3$-dimensional Bessel process and the general models. By decomposing the hitting times of consecutive sites into loops, we obtain identities, called loop identities, for the generating functions of the hitting times. After proving this decomposition both combinatorially and inductively, we consider the case that sites are equally distributed. Then, from loop identities, we derive expressions of Bernoulli and Euler polynomials, in terms of Euler polynomials of higher-orders.