Power series with inverse binomial coefficients and harmonic numbers

基于半整数二项式系数的调和卡特兰数级数

Harmonic Catalan Number Series via Half-Integer Binomial Coefficients

该研究通过半整数参数的交替二项式恒等式，结合广义二项式系数的显式表示与相关恒等式，推导出含卡特兰数、调和数的闭式求和公式，为转化恒等式提供了系统机制。

On the divisibility of sums of even powers of $q$-binomial coefficients

We prove the divisibility conjecture on sums of even powers of $q$-binomial coefficients, which was recently proposed by Guo, Schlosser and Zudilin. Our proof relies on two $q$-harmonic series congruences due to Shi and Pan.

二项式变换与序列的二项式卷积

Binomial Transforms and the Binomial Convolution of Sequences

本文研究了二项式变换与序列的二项式卷积之间的关系，推导了涉及多种数列的新恒等式，并提出构造新二项式变换对的方法。

无穷二项式和恒等式的发现与证明

Discovering and Proving Infinite Binomial Sums Identities

我们考虑无穷远处的二项和逆二项和，并将它们重写为一小组常数，例如$π$或$\log(2)$的幂。为了进行这些简化，我们将级数视为生成级数的特化。对于这些生成级数，我们导出了根值迭代积分的整表示。利用代换，我们将积分表示为分圆调和多项式。最后，利用已知的分圆调和多项式之间的关系，我们推导出几个常数的表达式。

一族包含广义调和数的求和公式

A family of summation formulas involving generalized harmonic numbers

将导数算子与伸缩法中的二项式和相结合，建立了一族包含广义调和数的求和公式。

Congruences on sums of $q$-binomial coefficients

We establish a $q$-analogue of Sun--Zhao's congruence on harmonic sums. Based on this $q$-congruence and a $q$-series identity, we prove a congruence conjecture on sums of central $q$-binomial coefficients, which was recently proposed by Guo. We also deduce a $q$-analogue of a congruence due to Apagodu and Zeilberger from Guo's $q$-congruence.

新的多重调和和恒等式

New Multiple Harmonic Sum Identities

考虑一类特殊的含有调和数的二项和，利用部分分式分解的初等方法证明了三个恒等式。给出了它在无穷级数和同余式中的一些应用。

伸缩方法、导数算子与调和数恒等式

Telescoping method, derivative operators and harmonic number identities

根据套叠法，给出了一个简单的二项式和。通过将导数算子应用于上述方程，我们建立了几个一般的调和数恒等式，包括一些已知的结果。

部分分式分解与调和数恒等式

Partial-Fraction Decompositions and Harmonic Number Identities

利用部分分式方法研究了有理函数的分解问题。我们将建立几个关于调和数和广义Aesthetic数的显著恒等式，包括与关于Aesthetic数的Beukers猜想相关的二项式调和数恒等式。

Logarithms of a binomial series: A Stirling number approach

The $p$-th power of the logarithm of the Catalan generating function is computed using the Stirling cycle numbers. Instead of Stirling numbers, one may write this generating function in terms of higher order harmonic numbers.

Euler型级数变换公式、函数的Hadamard积及调和数恒等式

Series Transformation Formulas of Euler Type, Hadamard Product of Functions, and Harmonic Number Identities

利用两个函数的Hadamard积的整表示证明了几个欧拉型级数变换公式。作为应用，我们得到三个二项式身份涉及调和数和身份的拉盖尔多项式。本文还对某些含调和数的幂级数级数进行了封闭形式的计算

涉及多重类调和数的组合恒等式

Combinatorial identities with multiple harmonic-like numbers

该研究利用生成函数方法推导多重类调和数相关二项式和的闭式，并给出多个推论、示例及与超调和数和奇调和数相关的组合恒等式。

递归二项式系数的生成函数族及其应用

A Family of Generating Functions for Reciprocal Binomial Coefficients and Its Applications

本文提出递归二项式系数的生成函数族，推导相关恒等式并展示其在无限序列评估中的应用。

包含中心二项式系数和高阶调和数的级数

Series involving central binomial coefficients and higher-order harmonic numbers

本文通过模性将包含中心二项式系数和高阶调和数的级数转化为狄利克雷L函数的特殊值，并推导了若干具体恒等式。

涉及二项式系数与特殊数的双重求和

Double sums involving binomial coefficients and special numbers

该研究提出一种初等方法处理内层为不完全二项式和的双重求和问题，并将核心恒等式应用于含调和数、Fibonacci数等多种特殊数的双重求和场景。

涉及两个二项式系数乘积的超同余式模 $p^4$

Supercongruences involving products of two binomial coefficients modulo $p^4$

本文证明了关于两个二项式系数乘积的超同余式模p⁴的猜想，涉及调和数和二项式系数的求和问题。

关于欧拉变换与取整函数

On the Euler transform and the floor function

本文推导了由取整函数加权的广义二项式和的欧拉变换表达式，并通过例子重新发现已知恒等式、证明新恒等式，得到涉及调和数、中心二项式系数和黎曼 zeta 函数的新的闭式结果。

Some Double Sums Involving Ratios of Binomial Coefficients Arising From Urn Models

In this paper we discuss a class of double sums involving ratios of binomial coefficients. The sums are of the form \[ \sum_{j=0}^{n} \sum_{i=0}^j \frac{\binom{f_1(n)}{i}}{\binom{f_2(n)}{j}}\,c^{i-j}, \] where $f_1, f_2$ are functions of $n$. Such sums appear in the analyses of the Mabinogion urn and the Ehrenfest urn in probability. Using hypergeometric functions, we are able to simplify these sums, and in some cases express them in terms of the harmonic numbers.

与二项式变换相关的二重和

Double sums associated with binomial transforms

本文针对内层为不完整二项式和的二重和展开研究，证明其与二项式变换对相关的新结果，推导了多类特殊数、多项式族相关的二重和恒等式及含双曲函数的新二重和。

复合逆，完备映射，正交拉丁方和Bent函数

Compositional inverses, complete mappings, orthogonal Latin squares and bent functions

我们研究置换多项式，完全映射，相互正交拉丁方和弯曲向量函数的合成逆。最近在[33]中得到了有限域上线性化置换二项式的合成逆。在[29]中还注意到，计算子空间的双射的逆可以应用于确定与线性化多项式相关的某些置换类的合成逆。本文给出了一类置换迹映射核的线性化二项式的合成逆。作为这个结果的应用，我们给出了一类完全映射的合成逆。这个完全映射类改进了[34]中给出的一个最近的构造。我们还递归地构造了一类涉及多迹函数的完全映射。最后利用这些完全映射导出了一组相互正交拉丁方，并从Maiorana-McFarland类构造了一类$p$ -元bent向量函数.

Some Combinatorial Identities some of which involving Harmonic Numbers

A product difference equation is proved and used for derivation by elementary methods of four combinatorial identities, eight combinatorial identities involving generalized harmonic numbers and eight combinatorial identities involving classical harmonic numbers. For the binomial coefficients the definition with gamma functions is used, thus also allowing non-integer arguments in the identities. The generalized harmonic numbers in this case are harmonic numbers with a complex offset, where the classical harmonic numbers are a special case with offset zero.

二项式系数与无符号斯特林数的矩阵乘积

Matrix products of binomial coefficients and unsigned Stirling numbers

本文研究二项式系数与无符号斯特林数构成的矩阵乘积和式，分析其闭式表达、组合解释、递推关系及多项式基变换系数意义。

涉及奇调和数的组合和、级数与积分

Combinatorial sums, series and integrals involving odd harmonic numbers

本文提出了涉及中心二项式系数、调和数和奇调和数的常生成函数，基于此计算了无穷级数的闭形式，并给出了Catalan数相关的组合和恒等式及Fibonacci数相关的对数积分分析。

New series with Cauchy and Stirling numbers, Part 2

We evaluate in closed form several series involving products of Cauchy numbers with other special numbers (harmonic, skew-harmonic, hyperharmonic, and central binomial). Similar results are obtained with series involving Stirling numbers of the first kind. We focus on several particular cases which give new closed forms for Euler sums of hyperharmonic numbers and products of hyperharmonic and harmonic numbers.

有限重调和q-级数的对偶性

Duality for Finite Multiple Harmonic q-Series

我们定义了两个有限的q-类似物的某些多重调和级数与任意数量的自由参数，并证明了这些q-类似物的身份，表示他们在涉及高斯二项式系数的多重嵌套和。这些恒等式的特殊情况--例如，所有参数都等于1--在文献中出现过。只有一个参数的特殊情况下减少到一个单位的除数生成函数，这已经收到了一些关注与排序理论中的问题。一般情况下，可以被视为一个对偶结果，让人联想到普通的多重zeta函数的对偶关系。

On some Binomial Coefficient Identities with Applications

We present a different proof of the following identity due to Munarini, which generalizes a curious binomial identity of Simons. \begin{align*} \sum_{k=0}^{n}\binomα{n-k}\binom{β+k}{k}x^k &=\sum_{k=0}^{n}(-1)^{n+k}\binom{β-α+n}{n-k}\binom{β+k}{k}(x+1)^k, \end{align*} where $n$ is a non-negative integer and $α$ and $β$ are complex numbers, which are not negative integers. Our approach is based on a particularly interesting combination of the Taylor theorem and the Wilf-Zeilberger algorithm. We also generalize a combinatorial identity due to Alzer and Kouba, and offer a new binomial sum identity…

Applications of derivative and difference operators on some sequences

In this study, depending on the upper and the lower indices of the hyperharmonic number $h_{n}^{(r)}$, nonlinear recurrence relations are obtained. It is shown that generalized harmonic number and hyperharmonic number can be obtained from derivatives of the binomial coefficients. Taking into account of difference and derivative operators, several identities of the harmonic and hyperharmonic numbers are given. Negative-ordered hyperharmonic number is defined and its alternative representations are given.

与某些多项式恒等式相关的有限和

Finite sums associated with some polynomial identities

本文提出一个通用框架，从特定多项式恒等式推导涉及二项式系数乘积、调和数及平方调和数的有限组合和，并应用于讨论近期著名多项式恒等式相关的组合和。

避免2个字母的有符号模式

Avoiding 2-letter signed patterns

设B_n是超八面体群，即n个字母上所有符号置换的集合，B_n（T）是B_n中避开符号模式集合T的所有符号置换的集合。本文给出了集合B_n（T）的所有基数，其中$T \subseteq B_2$ .这允许我们通过二项式系数、二项式系数、卡塔兰数和斐波那契数的逆来表示这些基数。

Proving two conjectural series for $ζ(7)$ and discovering more series for $ζ(7)$

We give a proof of two identities involving binomial sums at infinity conjectured by Z-W Sun. In order to prove these identities, we use a recently presented method i.e. we view the series as specializations of generating series and derive integral representations. Using substitutions, we express these integral representations in terms of cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive the results. These methods are implemented in the computer algebra package HarmonicSums.

终止超几何级数的多重逆形式

Multiplicate inverse forms of terminating hypergeometric series

Gould-Hsu的逆级数关系的乘法形式使得能够研究Chu-Vandermonde-Gaujiang的、Pfaff-Saalschütz的求和定理以及Hagen和Rothe的二项式卷积公式的对偶关系。由此建立了终止超几何级数的若干恒等式和互反关系。利用二重反演，我们分别建立了（3）\cite{ChuVanGauss}和（4）\cite{PfaffSaalsch}中Chu-Vandermonde-Gauelmann和Pfaff-Saalschütz求和定理的几个对偶公式.最后，从Hagen-Rothe的卷积恒等式出发，根据二重、三重和多重反演，导出了几个终止平衡超几何级数的恒等式和互反关系。

乘法图上的调和函数与无限随机路上的Pitman逆变换

Harmonic functions on multiplicative graphs and inverse Pitman transform on infinite random paths

我们引入并刻画了Littelmann路径上的中心概率分布。其次，我们建立了广义Pitmann变换的大数定律和中心极限定理。然后，我们研究了由有限维李代数表示的张量幂幂定义的乘法图上的调和函数。最后，我们证明了存在一个广义Pitman变换的逆定义几乎肯定的一组无限路径上的Weyl室，并解释它是如何计算的。

广义Stirling数的幂和恒等式

Power sum identities with generalized Stirling numbers

给出了几个组合恒等式，包括含复变数的第二类Stirling函数。这些恒等式还包括第一类Stirling数、二项式系数和调和数。

On a class of transformations of sequences of complex numbers

In this paper we consider a transformation $L_a$ of sequences of complex numbers. We find the inverse transformation of $L_a$ as well as the inverse of a related transformation $\tilde{L}_a$. We explore a connection to the binomial transform and significantly generalize a previously known result. We also obtain new relations among many classical hypergeometric orthogonal polynomials as well as new formulas for sums involving terminating hypergeometric series.

Semifinite harmonic functions on the direct product of graded graphs

Indecomposible semifinite harmonic functions on the direct product of graded graphs are classified. As a particular case, the full list of indecomposible traces for the infinite inverse symmetric semigroup is obtained.