Wenchang Chu
By means of partial fraction method, we investigate the decomposition of rational functions. Several striking identities on harmonic numbers and generalized Apery numbers will be established, including the binomial-harmonic number identity associated with Beukers' conjecture on Apery numbers.
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We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given.
In terms of the telescoping method, a simple binomial sum is given. By applying the derivative operators to the equation just mentioned, we establish several general harmonic number identities including some known results.
Two types of finite series of products of harmonic numbers involving nonnegative integer powers are evaluated, also yielding two other important harmonic number identities. The recursion formulas for these sums are derived, which are easily translated into a computer program.
Some finite series of harmonic numbers involving certain reciprocals are evaluated. Products of such reciprocals are expanded in a sum of the individual reciprocals, leading to a computer program. A list of examples is provided.
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。
The integral representation of the Hadamard product of two functions is used to prove several Euler-type series transformation formulas. As applications we obtain three binomial identities involving harmonic numbers and an identity for the Laguerre polynomials. We also evaluate in a closed form certain power series with harmonic numbers
Recently, Chen, Hou and Jin used both Abel's lemma on summation by parts and Zeilberger's algorithm to generate recurrence relations for definite summations. Meanwhile, they proposed the Abel-Gosper method to evaluate some indefinite sums involving harmonic numbers. In this paper, we use the Abel-Gosper method to prove an identity involving the generalized harmonic numbers. Special cases of this result reduce to many famous identities. In addition, we use both Abel's lemma and the WZ method to verify and to discover identities involving harmonic numbers. Many interesting examples are also pres…
Summation by parts is used to find the sum of a finite series of generalized harmonic numbers involving a specific polynomial or rational function. The Euler-Maclaurin formula for sums of powers is used to find the sums of some finite series of generalized harmonic numbers involving nonnegative integer powers, which can be used to evaluate the sums of the finite series of generalized harmonic numbers involving polynomials. Many examples and a computer program are provided.
We establish two binomial coefficient--generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of the author, they are used to establish modulo $p^k$ ($k>1$) congruences between truncated generalized hypergeometric series, and a function which extends Greene's hypergeometric function over finite fields to the $p$-adic setting. A specialization of one of these congruences is used to prove an outstanding conjecture of Rodriguez-Villegas which relates a trun…
By applying the derivative operators to Chu-Vandermonde convolution, several general harmonic number identities are established.
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.
The classical hypergeometric summation theorems are exploited to derive several striking identities on harmonic numbers including those discovered recently by Paule and Schneider (2003).
By applying the derivative operator to the corresponding hypergeometric form of a $q$-series transformation due to Andrews [1,Theorem 4], we establish a general harmonic number identity. As the special cases of it, several interesting Chu-Donno type identities and Paule-Schneider type identities are displayed.
A generalization of the Chu-Vandermonde convolution is presented and proved with the integral representation method. This identity can be transformed into another identity, which has as special cases two known identities. Another identity that is closely related to this identity is presented and proved. Using the modified geometric series from another paper, some closely related identities are listed. Some corresponding harmonic number identities are derived, which have as special cases some known harmonic number identities. For one combinatorial sum a recursion formula is derived and used to …
On series identities involving $\binom{4k}k$ and harmonic numbers
本文证明了涉及二项式系数$\binom{4k}k$和调和数的多个级数恒等式，其中一些恒等式此前由Z.-W. Sun提出猜想。
We consider the reflection identities for harmonic sums at weight four. We decompose a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or positive values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible bilinear set of reflection identities at weight four which present the main result of the paper. We also discuss how other trilinear and quartic reflection identities can be easil…
The aim of this paper is to present a general algebraic identity. Applying this identity, we provide several formulas involving the q-binomial coefficients and the q-harmonic numbers. We also recover some known identities including an algebraic identity of D. Y. Zheng on q-Apéry numbers and we establish the q-analog of Euler's formula. The proposed results may have important applications in the theory of q-supercongruences.
In terms of the derivative operator and three hypergeometric series identities, several interesting summation formulas involving generalized harmonic numbers are established.
Identities and Generating Functions of Products of Generalized Fibonacci numbers, Catalan and Harmonic Numbers
本文研究了广义 Fibonacci 与 Lucas 数类的性质，获得了著名 Fibonacci 恒等式的类比及乘积卷积新恒等式，并写下了其乘积、多截、调和数与 Catalan 数的生成函数。
In this work we prove a new combinatorial identity and applying it we establish many finite harmonic sum identities. Among many others, we prove that \begin{equation*} \sum_{k=1}^{n}\frac{(-1)^{k-1}}{k}\binom{n}{k}H_{n-k}=H_n^2+\sum_{k=1}^{n}\frac{(-1)^{k}}{k^2\binom{n}{k}}, \end{equation*} and \begin{equation*} \sum_{k=1}^{n}\frac{(-1)^{k-1}}{k^2}\binom{n}{k}H_{n-k}=\frac{H_n[H_n^2+H_n^{(2)}]}{2}-\sum_{k=0}^{n-1}\frac{(-1)^k[H_n-H_k]}{(k+1)(n-k)\binom{n}{k}}. \end{equation*} Almost all of our results are new, while a few of them recapture know results.
We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $π,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as specializations of generating series. For these generating series, we derive integral representations in terms of root-valued iterated integrals or directly in terms of cyclotomic harmonic polylogarithms. Using substitutions, we express the root-valued iterated integrals as cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclo…
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。
In terms of the derivative operator and Watson-type $_3F_2$-series identities, three families of summation formulae involving generalized harmonic numbers are established.
We introduce a class of $f(t)$-factorials, or $f(t)$-Pochhammer symbols, that includes many, if not most, well-known factorial and multiple factorial function variants as special cases. We consider the combinatorial properties of the corresponding generalized classes of Stirling numbers of the first kind which arise as the coefficients of the symbolic polynomial expansions of these $f$-factorial functions. The combinatorial properties of these more general parameterized Stirling number triangles we prove within the article include analogs to known expansions of the ordinary Stirling numbers by…
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。
By means of the derivative operator and Whipple-type $_3F_2$-series identities, two families of summation formulae involving generalized harmonic numbers are established.
In the framework of nonstandard analysis, Bang-He Li and the author defined the product of any two distributions on $R^n$ via their harmonic representations. The product of $δ(x_{1},..., x_{n})$ and $δ(x_{1})$ was calculated by Kuribayashi and the author in [LK]. In this paper, the result of [LK] is improved to $$δ(x_{1},..., x_{n})\circ δ(x_{1}) =\dfrac{1}{2πρ} δ(x_{1},..., x_{n}) {mod} {infinitesimals}$$ where $ρ$ is a positive infinitesimal. Moreover two combinatorial identities are obtained as byproducts.
We formulate several open questions related to enumerative combinatorics, which arise in the spectral analysis of Hill operators with trigonometric polynomial potentials.
Using basic hypergeometric functions and partial fraction decomposition we give a new kind of generalization of identities due to Uchimura, Dilcher, Van Hamme, Prodinger, and Chen-Fu related to divisor functions. An identity relating Lambert series to Eulerian polynomials is also proved.
Osburn and Schneider derived several combinatorial identities involving harmonic numbers using the computer programm Sigma. Here, they are derived by partial fraction decomposition and creative telescoping.
Additive decompositions of multiplicative subgroups via differential identities
本文针对素域真乘法子群的加性分解分类问题，在 Kalmynin 等人的通用框架基础上，给出了一种简化论证的全新自包含证明，明确唯一非平凡例子是阶为4的子群。
In this paper, we present a formula for generating various exotic series in the spirit of Ovidiu Furdui and Alina Sintamarian. Our new series (evaluated in closed form) involve Bernoulli, harmonic, and Catalan numbers. Also Stirling numbers of the second kind, other special numbers, and exponential polynomials. The results include series identities with Laguerre polynomials and derangement polynomials.
We define two finite q-analogs of certain multiple harmonic series with an arbitrary number of free parameters, and prove identities for these q-analogs, expressing them in terms of multiply nested sums involving the Gaussian binomial coefficients. Special cases of these identities--for example, with all parameters equal to 1--have occurred in the literature. The special case with only one parameter reduces to an identity for the divisor generating function, which has received some attention in connection with problems in sorting theory. The general case can be viewed as a duality result, remi…
We decompose the $\hat{\mathfrak{sl}}(n)$-module $V(Λ_0) \otimes V(Λ_0)$ and give generating function identities for the outer multiplicities. In the process we discover some seemingly new partition identities in the cases $n=2,3$.
Several combinatorial identities are presented, involving Stirling functions of the second kind with a complex variable. The identities involve also Stirling numbers of the first kind, binomial coefficients and harmonic numbers.
Some notes on an identity of Frisch
本文利用 Frisch 的组合恒等式证明并推广了涉及调和数的若干经典恒等式，并进一步推导出关于奇调和数的新组合恒等式。
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.
In this paper we obtain the LU-decomposition of a noncommutative linear system of equations that, in the rank one case, characterizes the image of the Lepowsky homomorphism $U(\lieg)^{K}\to U(\liek)^{M}\otimes U(\liea)$. This LU-decomposition can be transformed into very simple matrix identities, where the entries of the matrices involved belong to a special class of Jacobi polynomials. In particular, each entry of the L part of the original system is expressed in terms of a single ultraspherical Jacobi polynomial. In turns, these matrix identities yield a biorthogonality relation between the …
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。
By applying p-adic integral on the set of p-adic integers in [27] (Interpolation Functions for New Classes Special Numbers and Polynomials via Applications of p-adic Integrals and Derivative Operator, Montes Taurus J. Pure Appl. Math. 3 (1), ...--..., 2021 Article ID: MTJPAM-D-20-00000), we constructed generating function for the special numbers and polynomials involving the following combinatorial sum and numbers: y(n,λ)=\sum_{j=0}^{n}\frac{(-1)^{n}}{(j+1)λ^{j+1}\left(λ-1\right) ^{n+1-j}} The aim of this paper is to use the numbers y(n,λ) to derive some new and novel identities and formulas a…
We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial coefficients. This approach can be used to verify and discover identities involving harmonic numbers and derangement numbers. As examples, we use the Abel-Zeilberger algorithm to prove the Paule-Schneider identities, the Apery-Schmidt-Strehl identity, Calkin's identity and some identities involving Fibonacci numbers.
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