On a multiple harmonic power series

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.

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.

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.

A special constant and series with zeta values and harmonic numbers

In this paper we demonstrate the importance of a mathematical constant which is the value of several interesting numerical series involving harmonic numbers, zeta values, and logarithms. We also evaluate in closed form a number of numerical and power series.

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.

受拉马努金启发、涉及调和数的 $1/π$ 级数

Ramanujan-inspired series for $1/π$ involving harmonic numbers

本文利用导数算子作用于超几何级数或WZ对中的已知恒等式，构造出七个含调和数的级数，其中六个为关于 $1/π$ 的类拉马努金公式，并由此证明了孙的五个猜想级数。

Power series with skew-harmonic numbers, dilogarithms, and double integrals

The skew-harmonic numbers are the partial sums of the alternating harmonic series, i.e. the expansion of log(2). We evaluate in closed form various power series and numerical series with skew-harmonic numbers. This provides a simultaneous solution of two recent problems by Ovidiu Furdui in the American Mathematical Monthly and the College Mathematics Journal. We also present and discuss representations involving the dilogarithm and the trilogaithm which are related to our results. Finally, we provide the evaluations of several double integrals in terms of classical constants.

Asymptotic formulas for harmonic series in terms of a non-trivial zero on the critical line

In this article, we develop two types of asymptotic formulas for harmonic series in terms of single non-trivial zeros of the Riemann zeta function on the critical line. The series is obtained by evaluating the complex magnitude of an alternating and non-alternating series representation of the Riemann zeta function. Consequently, if the asymptotic limit of the harmonic series is known, then we obtain the Euler-Mascheroni constant with $\log(k)$. We further numerically compute these series for different non-trivial zeros. We also investigate a recursive formula for non-trivial zeros.

A note on the sum of finite multiple harmonic $q$-series on $r\text{-}(r+1)$ indices

We study the sum of the finite multiple harmonic $q$-series on $r\text{-}(r+1)$ indices at roots of unity with $r=1,2,3$. And we give the equivalent conditions of two conjectures regarding cyclic sums of finite multiple harmonic $q$-series on $1\text{-}2\text{-}3$ indices at roots of unity, posed recently by Kh. Pilehrood, T. Pilehrood and R. Tauraso.

Special values of finite multiple harmonic q-series at roots of unity

We study special values of finite multiple harmonic q-series at roots of unity. These objects were recently introduced by the authors and it was shown that they have connections to finite and symmetric multiple zeta values and the Kaneko-Zagier conjecture. In this note we give new explicit evaluations for finite multiple harmonic q-series at roots of unity and prove Ohno-Zagier-type relations for them.

Fractal Subseries of the Harmonic Series

We study the convergence of certain subseries of the harmonic series corresponding to increasing sequences of integers whose digits in a certain base are not uniformly distributed. We also discuss the case of irregular sequences, where the frequency distribution of some of the digits does not exist. Examples are given for irregular sequences where the corresponding harmonic subseries is convergent, or divergent, respectively.

Derivations on the algebra of multiple harmonic q-series and their applications

We introduce derivations on the algebra of multiple harmonic q-series and show that they generate linear relations among the q-series which contain the derivation relations for a q-analogue of multiple zeta values due to Bradley. As a byproduct we obtain Ohno-type relations for finite multiple harmonic q-series at a root of unity.

A generalization of the duality for finite multiple harmonic q-series

Recently, Bradley studied partial sums of multiple q-zeta values and proved a duality result. In this paper, we present a generalization of his result.

新的调和数级数

New harmonic number series

基于psi函数的新表示推导出涉及调和数与逆阶乘级数的新闭式，并以更简单的方法重新发现了Borwein等人1995年提出的著名二次Euler和。

A Note on Harmonic number identities, Stirling series and multiple zeta values

We study a general type of series and relate special cases of it to Stirling series, infinite series discussed by Choi and Hoffman, and also to special values of the Arakawa-Kaneko zeta function, complementing and generalizing earlier results. Moreover, we survey properties of certain truncated multiple zeta and zeta star values, pointing out their relation to finite sums of harmonic numbers. We also discuss the duality result of Hoffman, relating binomial sums and truncated multiple zeta star values.

关于涉及多重$t$-调和星和的Apéry型级数的评估

On the Evaluation of Apéry-Like Series Involving Multiple $t$-Harmonic Star Sums

本文通过初等方法评估了涉及多重$t$-调和星和的Apéry型级数，得出了显式的闭式表达式，并给出了若干具体例子。

On generalized harmonic numbers, Tornheim double series and linear Euler sums

Direct links between generalized harmonic numbers, linear Euler sums and Tornheim double series are established in a more perspicuous manner than is found in existing literature. We show that every linear Euler sum can be decomposed into a linear combination of Tornheim double series of the same weight. New closed form evaluations of various Euler sums are presented. Finally certain combinations of linear Euler sums that are reducible to Riemann zeta values are discovered.

Almost holomorphic Poincare series corresponding to products of harmonic Siegel-Maass forms

We investigate Poincaré series, where we average products of terms of Fourier series of real-analytic Siegel modular forms. There are some (trivial) special cases for which the products of terms of Fourier series of elliptic modular forms and harmonic Maass forms are almost holomorphic, in which case the corresponding Poincaré series are almost holomorphic as well. In general this is not the case. The main point of this paper is the study of Siegel-Poincaré series of degree $2$ attached to products of terms of Fourier series of harmonic Siegel-Maass forms and holomorphic Siegel modular forms. …

Multiple Harmonic Series I: Generalizations of Wolstenholme's Theorem

In this paper we will study the p-divisibility of partial sums of multiple zeta value series. In particular we provide some generalizations of the classical Wolstenholme's Theorem.

正交艾森斯坦级数在调和点处的性质与奇异权模形式

Orthogonal Eisenstein Series at Harmonic Points and Modular Forms of Singular Weight

本文利用theta提升研究正交非全纯Eisenstein级数在调和点的行为，证明奇异权下其在$s=1$处的留数生成全纯正交模形式，并给出构造满射的充分条件。

Tornheim-like series, harmonic numbers and zeta values

Explicit evaluations of the Tornheim-like double series in the form \[ \sum_{n,m=1}^\infty \frac{H_{n+m+s}}{nm\left( n+m+s \right)},\ s\in \mathbb{N\cup } \left\{ 0 \right\} \] and their extensions are given. Furthermore, series of the type \[ \sum_{m=1}^\infty \frac{2H_{2m+1}-H_{m}}{2m\left( 2m+1 \right)} \] and some other Tornheim-like multiple series are evaluated in terms of the zeta values.

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.

单位根上有限多重调和q级数的广濑关系的类似物

An analogue of Hirose's relation for finite multiple harmonic $q$-series at roots of unity

本文建立单位根上有限多重调和q级数的广濑关系类似物，并将其用于证明Hessami Pilehrood等人提出的循环和猜想。

H-Harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Some Indefinite Theta Series

It was shown in previous work that the one-variable $\widehatμ$-function defined by Zwegers (and Zagier) and his indefinite theta series attached to lattices of signature $(r\!+\!1,1)$ are both Heisenberg harmonic Maaß-Jacobi forms. We extend the concept of Heisenberg harmonicity to Maaß-Jacobi forms of arbitrary many elliptic variables, and produce indefinite theta series of "product type" for non-degenerate lattices of signature $(r\!+\!s,s)$. We thus obtain a clean generalization of $\widehatμ$ to these negative definite lattices. From restrictions to torsion points of Heisenberg harmonic M…

Finite and symmetric colored multiple zeta values and multiple harmonic q-series at roots of unity

The Kaneko-Zagier conjecture states that finite and symmetric multiple zeta values satisfy the same relations. In the previous work with H.~Bachmann and Y.~Takeyama, we proved that the finite and symmetric multiple zeta value are obtained as an `algebraic' and `analytic' limit at $q\rightarrow 1$ of a certain truncated multiple harmonic $q$-series, and studied its relations in order to give partial evidence of the Kaneko-Zagier conjecture. In this paper, we start with truncated multiple harmonic $q$-series of level $N$, which is a $q$-analogue of the truncated colored multiple zeta value. We i…

Polynomials Related to Harmonic Numbers and Evaluation of Harmonic Number Series II

In this paper we focus on r-geometric polynomials, r-exponential polynomials and their harmonic versions. It is shown that harmonic versions of these polynomials and their generalizations are useful to obtain closed forms of some series related to harmonic numbers.

Apéry型级数与彩色多重Zeta值

Apéry-Type Series and Colored Multiple Zeta Values

本文结合迭代积分与傅里叶-勒让德级数展开，研究含多重t-调和和的新型Apéry型级数，推广旧级数类并证明其可表为4级彩色多重Zeta值Q-线性组合的实部或虚部。

A q-analogue of Wolstenholme's harmonic series congruence

We establish a q-analogue of Wolstenholme's harmonic series congruence.

Polynomials Related to Harmonic Numbers and Evaluation of Harmonic Number Series I

In this paper we focus on two new families of polynomials which are connected with exponential polynomials and geometric polynomials. We discuss their generalizations and show that these new families of polynomials and their generalizations are useful to obtain closed forms of some series related to harmonic numbers.

关于涉及调和数幂的和

On sums involving powers of harmonic numbers

本文研究由调和数幂生成的Dirichlet级数，推导相关级数并分析其解析性质与渐近行为，进而揭示其与Euler-Mascheroni常数无理性证明之间的可能联系。

Watson-type 3F2-series and summation formulae involving generalized harmonic numbers

In terms of the derivative operator and Watson-type $_3F_2$-series identities, three families of summation formulae involving generalized harmonic numbers are established.

Power series with inverse binomial coefficients and harmonic numbers

We construct the generating function for products of inverse central binomial coefficients with harmonic numbers.

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

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

Whipple-type $_3F_2$-series and summation formulae involving generalized harmonic numbers

By means of the derivative operator and Whipple-type $_3F_2$-series identities, two families of summation formulae involving generalized harmonic numbers are established.

Exotic series with Bernoulli, harmonic, Catalan, and Stirling numbers

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.

椭圆调和级数再探

Ellipsephic harmonic series revisited

该研究针对椭圆（类肯普纳）调和级数，推广了文献中部分结果，给出了任意底数下任意数字或数字块对应的级数的完整结论。

An extension of Boyd's $p$-adic algorithm for the harmonic series

In this paper we will extend a $p$-adic algorithm of Boyd in order to study the size of the set: \[J_p(y)=\left\{n :\sum_{j=1}^{n}\frac{y^j}{j}\equiv 0(\mod p)\right\}.\] Suppose that $p$ is one of the first 100 odd primes and $y\in\{1,2,...,p-1\}$, then our calculations prove that $|J_p(y)|<\infty$ in 24240 out of 24578 possible cases. Among other results we show that $|J_{13}(9)|=18763$. The paper concludes by discussing some possible applications of our method to sums involving Fibonacci numbers.

On $3-2-1$ values of finite multiple harmonic $q$-series at roots of unity

We mainly answer two open questions about finite multiple harmonic $q$-series on 3-2-1 indices at roots of unity, posed recently by H. Bachmann, Y. Takeyama, and K. Tasaka. Two conjectures regarding cyclic sums which generalize the given results are also provided.

求和项涉及调和数的级数

Series with summands involving harmonic numbers

本文研究了涉及不超过四阶调和数的级数，发现了部分级数的精确值，并提出70个相关猜想。

利用奇数调和级数近似对数、阶乘和欧拉-马歇罗尼常数

Approximation Of Logarithm, Factorial And Euler Mascheroni Constant Using Odd Harmonic Series

本文通过奇数调和级数推导出对数、阶乘和欧拉-马歇罗尼常数的近似公式

An Analogue of Weil's Converse Theorem for Harmonic Maass Forms of Polynomial Growth

We construct a family of harmonic Maass forms of polynomial growth of any level corresponding to any cusp whose shadows are Eisenstein series of integral weight. We further consider Dirichlet series attached to a harmonic Maass form of polynomial growth, study its analytic properties, and prove an analogue of Weil's converse theorem.

Harmonic theta series and the Kodaira dimension of $\mathcal{A}_6$

We construct a basis of the space ${\text S}_{14}({\text{Sp}}_{12}({\mathbb Z}))$ of Siegel cusp forms of degree $6$ and weight $14$ consisting of harmonic theta series. One of these functions has vanishing order $2$ at the boundary which implies that the Kodaira dimension of $\mathcal{A}_6$ is non-negative.

Linear Relations Among Poincare Series via Harmonic Weak Maass Forms

We discuss the problem of the vanishing of Poincaré series. This problem is known to be related to the existence of weakly holomorphic forms with prescribed principal part. The obstruction to the existence is related to the pseudomodularity of Ramanujan's mock theta functions. We embed the space of weakly holomorphic modular forms into the larger space of harmonic weak Maass forms. From this perspective we discuss the linear relations between Poincaré series and the connection to Ramanujan's mock theta functions.

与调和数、斐波那契数及中心二项式系数 $\binom{2n}{n}$ 相关的级数

Series Associated with Harmonic Numbers, Fibonacci Numbers and Central Binomial Coefficients $\binom{2n}{n}$

本文通过直接变换而非传统超几何方法，发现了涉及中心二项式系数、调和数与斐波那契数的多种级数，并给出了 $\zeta(2)$ 的新级数表示。

The Algebra of a q-Analogue of Multiple Harmonic Series

We introduce an algebra which describes the multiplication structure of a family of q-series containing a q-analogue of multiple zeta values. The double shuffle relations are formulated in our framework. They contain a q-analogue of Hoffman's identity for multiple zeta values. We also discuss the dimension of the space spanned by the linear relations realized in our algebra.

通过Wilf-Zeilberger种子发现包含调和数的超几何级数

Discovering hypergeometric series with harmonic numbers via Wilf-Zeilberger seeds

本文通过Wilf-Zeilberger种子方法发现包含调和数的超几何级数，并首次获得深度为二的多重ζ值ζ(5,3)的快速收敛级数，同时推测WZ-种子背后的格雷德代数结构。

Some Generalized Harmonic Number Identities

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.

Quasi-symmetric functions and mod p multiple harmonic sums

We present a number of results about (finite) multiple harmonic sums modulo a prime, which provide interesting parallels to known results about multiple zeta values (i.e., infinite multiple harmonic series). In particular, we prove a "duality" result for mod p multiple harmonic sums similar to (but distinct from) that for multiple zeta values. We also exploit the Hopf algebra structure of the quasi-symmetric functions to do calculations with multiple harmonic sums mod p, and obtain, for each weight through 9, a set of generators for the space of weight-n multiple harmonic sums mod p. When comb…

New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner's series

In this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences $(\{1\}^a,c,\{1\}^b),$ $(\{2\}^a,c,\{2\}^b)$ and prove a number of congruences for these sums modulo a prime $p.$ The congruences obtained allow us to find nice $p$-analogues of Leshchiner's series for zeta values and to refine a result due to M. Hoffman and J. Zhao about the set of generators of the multiple harmonic sums of weight 7 and 9 modulo $p$. Moreover, we are also able to provide a new proof of Zagier's formula for $ζ^{*}(\{2\}^a,3,\{2\}^b)$ based on a finite…

L-values of harmonic Maass forms

Bruinier, Funke, and Imamoglu have proved a formula for what can philosophically be called the "central $L$-value" of the modular $j$-invariant. Previously, this had been heuristically suggested by Zagier. Here, we interpret this "$L$-value" as the value of an actual $L$-series, and extend it to all integral arguments and to a large class of harmonic Maass forms which includes all weakly holomorphic cusp forms. The context and relation to previously defined $L$-series for weakly holomorphic and harmonic Maass forms are discussed. These formulas suggest possible arithmetic or geometric meaning …

On a Double Series Representation of the Natural Logarithm, the Asymptotic Behavior of Hölder Means, and an Elementary Estimate for the Prime Counting Function

We present many novel results in number theory, including a double series formula for the natural logarithm and a proof concerning the Hölder mean based on the functional equation for the Riemann zeta function. We find a harmonic mean analogue of Chebyshev's inequality for the prime counting function involving the Euler-Mascheroni constant. Furthermore, we define a function taking the Hölder mean of all positive integers up to a given number and investigate its asymptotic behavior, finding two different patterns which are separated by the harmonic mean. Additionally, we discuss the behavior of…

On connection between values of Riemann zeta function at integers and generalized harmonic numbers

Using Euler transformation of series we relate values of Hurwitz zeta function at integer and rational values of arguments to certain rapidly converging series where some generalized harmonic numbers appear. The form of these generalized harmonic numbers carries information about the values of the arguments of Hurwitz function. In particular we prove: $\forall k\in \mathbb{N}:$ $ζ(k,1)\allowbreak =\allowbreak \frac{2^{k-1}}{2^{k-1}-1}% \sum_{n=1}^{\infty }\frac{H_{n}^{(k-1)}}{n2^{n}},$ where $H_{n}^{(k)}$ are defined below generalized harmonic numbers. Further we find generating function of th…

Series with Hermite Polynomials and Harmonic Numbers

This paper is now part of the new paper "Series with Hermite polynomials and applications" arXiv:1710.00687.

k秩泰勒族：典范调和提升、主部与拉德马赫级数

The k-Rank Taylor Family: Canonical Harmonic Lifts, Principal Parts, and Rademacher Series

该研究针对k≥2的Garvan修正k秩矩，构造k秩泰勒族的典范调和提升，推导其主部、拉德马赫展开及精确递归，k=2时得到2标记Durfee符号数的精确公式。