Some binomial series obtained by the WZ-method

Evaluation of binomial series with harmonic numbers

We define a special function related to the digamma function and use it to evaluate in closed form various series involving binomial coefficients and harmonic numbers.

Sums of series involving central binomial coefficients & harmonic numbers

This paper contains a number of series whose coefficients are products of central binomial coefficients & harmonic numbers. An elegant sum involving $ζ(2)$ and two other nice sums appear in the last section.

Dirichlet series and series with Stirling numbers

This paper presents a number of identities for Dirichlet series and series with Stirling numbers of the first kind. As coefficients for the Dirichlet series we use Cauchy numbers of the first and second kinds, hyperharmonic numbers, derangement numbers, binomial coefficients, central binomial coefficients, and Catalan numbers.

Power Series with Binomial Sums and Asymptotic Expansions

This paper is a study of power series, where the coefficients are binomial expressions (iterated finite differences). Our results can be used for series summation, for series transformation, or for asymptotic expansions involving Stirling numbers of the second kind. In certain cases we obtain asymptotic expansions involving Bernoulli polynomials, poly-Bernoulli polynomials, or Euler polynomials. We also discuss connections to Euler series transformations and other series transformation formulas.

On four families of power series involving harmonic numbers and central binomial coefficients

We present several sequences involving harmonic numbers and the central binomial coefficients. The calculational technique is consists of a special summation method that allows, based on proper two-valued integer functions, to calculate different families of power series which involve odd harmonic numbers and central binomial coefficients. Furthermore it is shown that based on these series a new type of nonlinear Euler sums that involve odd harmonic numbers can be calculated in terms of zeta functions.

Hurwitz-Lerch 型中心二项式级数

Hurwitz-Lerch type central binomial series

本文通过添加一个实参数定义了中心二项式级数的 Hurwitz 版本，并研究了其在整数点处的值。

一些涉及二项式系数 $C(4n,2n)$ 的交错级数的求值

Evaluation of some alternating series involving the binomial coefficients $C(4n,2n)$

本文研究分母含中心二项式系数的交错无穷级数，给出多个闭式求值，并推广了 Sprugnoli 于2006年实验发现的恒等式。

Discovering and Proving Infinite Binomial Sums Identities

We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $π$ or $\log(2)$. 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. Using substitutions, we express the interated integrals as cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive expressions in terms of several constants.

On Rational Invariant Summation of p-Adic Power Series with Binomial Coefficient

In the work we have considered p-adic functional series with binomial coefficients and discussed its p-adic convergence. Then we have derived a recurrence relation following with a summation formula which is invariant for rational argument. More precisely, we have investigated a sufficient condition under which the p-adic power series converges to a rational invariant sum 0. Finally we have shown application of the invariant summation formula to get some intersting relations involving Bernoulli numbers and Bernoulli polynomials.

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.

A hypergeometric proof for a binomial identity related to $1/π$

We show that a binomial identity arising in the context of the study of series expansions of $1/π$ can be seen as an incarnation of Whipples second theorem for hypergeometric series.

新的中心 $q$-二项式恒等式

New central $q$-binomial identities

该研究受Vignat及其中一位作者的早期工作启发，建立了若干涉及中心 $q$-二项式系数的新级数求值结果。

A new family of series expansions for $1/π$ and a binomial identity

A doubly infinite set of series expansion for $1/π$ are reported. They follow trivially from a formal expansion for the quotient of the values taken by the gamma function for two (complex) arguments differing by an integer plus one half, obtained by an alternative computation of the Wronskian of the modified Bessel functions. The same formal expansion allows to discover also a new binomial identity.

涉及中心二项式系数的更多级数类

Further Classes of Series Involving Central Binomial Coefficients

本文从依赖正整数参数的中心二项式系数无穷级数出发，利用指数Bell多项式、积分表示以及gamma与polygamma函数，将已知恒等式推广至复数参数并进一步扩展了相关结果。

On polynomials of binomial type, Ramanujan-Soldner constant and inverse logarithmic derivative operator

In this work we introduce interesting infinite series, related to Ramanujan-Soldner constant. Our method uses general properties of polynomials of binomial type and Lagrange inversion theorem. Also we study properties of the operator 1/dlog, acting on formal power series. In addition, several properties of polynomials of binomial type associated to elementary functions are discussed.

关于涉及$ζ(2n)$和二项式系数的若干有理 zeta 级数

On some rational zeta series involving $ζ(2n)$ and binomial coefficients

本文利用余切函数的导数多项式，给出了一类涉及$ζ(2n)$的有理 zeta 级数用 Hurwitz zeta 值表示的精确公式，并推广了作者此前的两个结果。

包含二项式系数的新级数（二）

New series involving binomial coefficients (II)

本文研究了包含二项式系数的新级数，通过具体例子证明了若干级数的闭合形式，并提出了新的猜想级数恒等式。

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.

通过分圆多重 zeta 值对逆二项式级数的新求值

New Evaluations of Inverse Binomial Series via Cyclotomic Multiple Zeta Values

本文利用 Au 基于分圆多重 zeta 值的求值方法，解决了 Chen 2010 年分析教材中关于逆二项式级数的公开问题。

轮廓积分与涉及广义二项式系数的 cyclotomic 阿佩里型级数

Contour Integration and Cyclotomic Apéry-Like Series Involving Generalized Binomial Coefficients

本文通过轮廓积分方法研究了一类涉及广义二项式系数的 cyclotomic 阿佩里型级数，推导出多重 polylogarithms 等表达式，并恢复了一些已知结果。

The Ongoing Binomial Revolution

The Binomial Theorem has long been essential in mathematics. In one form or another it was known to the ancients and, in the hands of Leibniz, Newton, Euler, Galois, and others, it became an essential tool in both algebra and analysis. Indeed, Newton early on developed certain binomial series (see Section \ref{newton}) which played a role in his subsequent work on the calculus. From the work of Leibniz, Galois, Frobenius, and many others, we know of its essential role in algebra. In this paper we rapidly trace the history of the Binomial Theorem, binomial series, and binomial coefficients, wit…

Some $n$-space $q$-binomial theorem extensions and similar identities

We give an $n$-space generalized $q$-binomial theorem, and some new $q$ series identities that resemble the traditional $q$ series partition generating functions. These identities enumerate stepping stone weighted vector partitions.

On sums related to central binomial and trinomial coefficients

A generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$ with $b,c\in\mathbb Z$. In this paper we investigate congruences and series for sums of terms related to central binomial coefficients and generalized central trinomial coefficients. The paper contains many conjectures on congruences related to representations of primes by certain binary quadratic forms, and 62 proposed new series for $1/π$ motivated by congruences and related dualities.

参数化Apéry型级数与Hurwitz型多重zeta值

Parametric Apéry-type Series and Hurwitz-type Multiple Zeta Values

本文通过迭代积分方法，将Apéry型级数的研究从中心二项式系数推广到参数化二项式系数，建立了其与Hurwitz型多重zeta值之间的显式关系。

The role of binomial type sequences in determination identities for Bell polynomials

Our paper deals about identities involving Bell polynomials. Some identities on Bell polynomials derived using generating function and successive derivatives of binomial type sequences. We give some relations between Bell polynomials and binomial type sequences in first part, and, we generalize the results obtained in [4] in second part.

Tiling proofs of Jacobi triple product and Rogers-Ramanujan identities

We use the method of tiling to give elementary combinatorial proofs of some celebrated $q$-series identities, such as Jacobi triple product identity, Rogers-Ramanujan identities, and some identities of Rogers. We give a tiling proof of the q-binomial theorem and a tiling interpretation of the q-binomial coefficients. A new generalized $ k $-product $q$-series identity is also obtained by employing the `tiling-method', wherein the generating function of the set of all possible tilings of a rectangular board is computed in two different ways to obtain the desired $q$-series identity. Several new…

Congruences involving the reciprocals of central binomial coefficients

We present several congruences modulo a power of prime $p$ concerning sums of the following type $\sum_{k=1}^{p-1}{m^k\over k^r}{2k\choose k}^{-1}$ which reveal some interesting connections with the analogous infinite series.

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。

On b-ary binomial coefficients with negative entries

Abstract. We generalize the $b$-ary binomial coefficients with negative entries, which is based on the generating function obtained in early work. Besides an explicit expression involving the restricted partition, several properties such as symmetry, congruence and Pascal-like recurrence are studies. Finally, we also provide two different generalizations, partially satisfying Pascal-like recurrences.

On some divisibility properties of binomial sums

In this paper, we consider two particular binomial sums \begin{align*} \sum_{k=0}^{n-1}(20k^2+8k+1){\binom{2k}{k}}^5 (-4096)^{n-k-1} \end{align*} and \begin{align*} \sum_{k=0}^{n-1}(120k^2+34k+3){\binom{2k}{k}}^4\binom{4k}{2k} 65536^{n-k-1}, \end{align*} which are inspired by two series for $\frac{1}{π^2}$ obtained by Guillera. We consider their divisibility properties and prove that they are divisible by $2n^2 \binom{2n}{n}^2$ for all integer $n\geq 2$. These divisibility properties are stronger than those divisibility results found by He, who proved the above two sums are divisible by $2n \b…

包含二项式系数的新级数（III）

New series involving binomial coefficients (III)

本文提出了一种对偶原理，用于发现26个新的无理数级数恒等式，涉及包含二项式系数的级数评估。

Whipple求和公式的双重和模拟式及多重二项和的受限求和公式

A double-sum analogue of Whipple's summation formula and a restricted sum formula for multiple binomial sums

该研究证明了Whipple的{}_3F_2求和公式的双重和模拟式，并将其应用得到Igarashi参数化多重zeta级数特殊情形下多重二项和的受限求和公式

Bi-periodic Fibonacci matrix polynomial and its binomial transforms

In this paper, we consider the matrix polynomial obtained by using bi-periodic Fibonacci matrix polynomial. Then, we give some properties and binomial transforms of the new matrix polynomials.

New series identities with Cauchy, Stirling, and harmonic numbers, and Laguerre polynomials

In this article we use an interplay between Newton series and binomial formulas in order to generate a number of series identities involving Cauchy numbers, harmonic numbers, Laguerre polynomials, and Stirling numbers of the first kind.

Supercongruences related to ${}_3F_2(1)$ involving harmonic numbers

We show various supercongruences for truncated series which involve central binomial coefficients and harmonic numbers. The corresponding infinite series are also evaluated.

On the operations of sequences in rings and binomial type sequences

Given a commutative ring with identity $R$, many different and interesting operations can be defined over the set $H_R$ of sequences of elements in $R$. These operations can also give $H_R$ the structure of a ring. We study some of these operations, focusing on the binomial convolution product and the operation induced by the composition of exponential generating functions. We provide new relations between these operations and their invertible elements. We also study automorphisms of the Hurwitz series ring, highlighting that some well--known transforms of sequences (such as the Stirling trans…

二项式系数幂和的Apéry极限与Franel的猜想

Sums of powers of binomials, their Apéry limits, and Franel's suspicions

本文显式确定了二项式系数幂和的Apéry极限，并在假设递归可通过创造性望远镜法获得的前提下，证明了关于这些序列递归阶的Franel猜想的最小阶版本。

交错Apéry型级数与八级彩色多重zeta值

Alternating Apéry-Type Series and Colored Multiple Zeta Values of Level Eight

本文研究交错Apéry型级数，证明其求值需用八级彩色多重zeta值，方法是以双曲三角函数1-形式替代普通三角函数。

Lehmer对与二项式级数

Lehmer pairs and binomial series

本文通过广义Hardy函数与二项式级数的关系，建立了cosθ(t)零点与Hardy函数零点之间的联系。

关于Sun型二项 harmonic 和的若干注记

Notes on certain binomial harmonic sums of Sun's type

本文证明并推广了Sun关于包含调和数和二项式系数乘积的无穷级数的猜想，通过自守对象方法求得闭合形式。

Some irreducibility results for truncated binomial expansions

For positive integers $n>k$, let $P_{n,k}(x)=\displaystyle\sum_{j=0}^k \binom{n}{j}x^j $ be the polynomial obtained by truncating the binomial expansion of $(1+x)^n$ at the $k^{th}$ stage. These polynomials arose in the investigation of Schubert calculus in Grassmannians. In this paper, the authors prove the irreducibility of $P_{n,k}(x)$ over the field of rational numbers when $2\leqslant 2k \leqslant n<(k+1)^3$.

关于第11级Ramanujan-Sato级数的一个问题

On a problem on Ramanujan-Sato series of level 11

本文解决了关于第11级Ramanujan-Sato级数系数$c_{11}(n)$的长期未解问题，通过引入并证明一个双二项式求和公式，直接类比较低级系数公式，成功给出了所有非负整数$n$的显式表达式。

多重阿佩里级数的积分表示

Integral Representations for Multiple Apéry-Like Series

本文通过初等方法获得多重阿佩里级数的积分表示，结合多重调和求和与中心二项式因子，利用分部积分与两个关键傅里叶展开推导出新恒等式。

Binomial sums related to rational approximations to $ζ(4)$

For the solution $\{u_n\}_{n=0}^\infty$ to the polynomial recursion $(n+1)^5u_{n+1}-3(2n+1)(3n^2+3n+1)(15n^2+15n+4)u_n -3n^3(3n-1)(3n+1)u_{n-1}=0$, where $n=1,2,...$, with the initial data $u_0=1$, $u_1=12$, we prove that all $u_n$ are integers. The numbers $u_n$, $n=0,1,2,...$, are denominators of rational approximations to $ζ(4)$ (see math.NT/0201024). We use Andrews's generalization of Whipple's transformation of a terminating ${}_7F_6(1)$-series and the method from math.NT/0311114.

Parametric binomial sums involving harmonic numbers

We present explicit formulas for the following family of parametric binomial sums involving harmonic numbers for $p=0,1,2$ and $|t|\leq1$. $$ \sum_{k=1}^{\infty}\frac{H_{k-1}t^k}{k^p\binom{n+k}{k}}\quad \mbox{and}\quad \sum_{k=1}^{\infty}\frac{t^k}{k^p\binom{n+k}{k}}. $$ We also generalize the following relation between the Stirling numbers of the first kind and the Riemann zeta function to polygamma function and give some applications. $$ ζ(n+1)=\sum_{k=n}^{\infty}\frac{s(k,n)}{kk!}, \quad n=1,2,3,... . $$ As examples, \begin{equation*} ζ(3)=\frac{1}{7}\sum_{k=1}^{\infty}\frac{H_{k-1}4^k}{k^2…

Some results about permutation properties of a kind of binomials over finite fields

Permutation polynomials have many applications in finite fields theory, coding theory, cryptography, combinatorial design, communication theory, and so on. Permutation binomials of the form $x^{r}(x^{q-1}+a)$ over $\mathbb{F}_{q^2}$ have been studied before, K. Li, L. Qu and X. Chen proved that they are permutation polynomials if and only if $r=1$ and $a^{q+1}\not=1$. In this paper, we consider the same binomial, but over finite fields $\mathbb{F}_{q^3}$ and $\mathbb{F}_{q^e}$. Two different kinds of methods are employed, and some partial results are obtained for them.