Ayhan Dil, Veli Kurt
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.
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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.
In the present paper, we introduce the concept of harmonic Tutte polynomials of matroids and discuss some of their properties. In particular, we generalize Greene's theorem, thereby expressing harmonic weight enumerators of codes as evaluations of harmonic Tutte polynomials.
Harmonic Geometric Polynomials via Geometric Polynomials and Their Applications
本文通过几何多项式构造调和几何多项式，并推导出其应用，包括调和数恒等式的推广和伯努利数的新表达式。
We establish a q-analogue of Wolstenholme's harmonic series congruence.
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 construct the generating function for products of inverse central binomial coefficients with harmonic numbers.
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
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 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.
Jacobi polynomials and harmonic weight enumerators of the first-order Reed--Muller codes and the extended Hamming codes
本文研究并给出了一阶Reed--Muller码与扩展Hamming码的调和重量枚举器及Jacobi多项式，并据此证明了这些码中不存在组合4-设计。
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.
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。
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。
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.
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-series of harmonic Maass forms and a summation formula for harmonic lifts
本文引入了与调和Maass形式相关的L-级数并证明其函数方程，建立了该L-级数的逆定理，并作为应用证明了给定尖点形式的调和提升全纯部分的求和公式。
In this short note, we will construct a harmonic Eisenstein series of weight one, whose image under the $ξ$-operator is a weight one Eisenstein series studied by Hecke.
The Newton series which interpolate finite multiple harmonic sums are useful in the study of multiple zeta values (MZV's). In this paper, we prove that these Newton series can be written as multiple series. As an application, we give a formula for MZV's which contains the duality.
In this paper, we give explicit evaluation for some infinite series involving generalized (alternating) harmonic numbers. In addition, some formulas for generalized (alternating) harmonic numbers will also be derived.
We define and study the interpolated finite multiple harmonic $q$-series. A generating function of the sums of the interpolated finite multiple harmonic $q$-series with fixed weight, depth and $i$-height is computed. Some Ohno-Zagier type relation with corollaries and some evaluation formulas of the interpolated finite multiple harmonic $q$-series at roots of unity are given.
We give a generalization of Wolstenholme's harmonic series congruence for the Lucas sequences.
Some numbers and polynomials related to degenerate harmonic and degenerate hyperharmonic numbers
本文引入退化调和-福比尼多项式及数，并推广至退化超调和情形，研究其性质、显式表达式和恒等式。
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.
We present a relationship between the generalized hyperharmonic numbers and the poly-Bernoulli polynomials, motivated from the connections between harmonic and Bernoulli numbers. This relationship yields numerous identities for the hyper-sums and several congruences.
The formulas discussed in the previous version are not new.
We give a new structural development of harmonic polynomials on Hamming space, and harmonic weight enumerators of binary linear codes, that parallels one approach to harmonic polynomials on Euclidean space and weighted theta functions of Euclidean lattices. Namely, we use the finite-dimensional representation theory of sl_2 to derive a decomposition theorem for the spaces of discrete homogeneous polynomials in terms of the spaces of discrete harmonic polynomials, and prove a generalized MacWilliams identity for harmonic weight enumerators. We then present several applications of harmonic weigh…
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.
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.
In recent years, some degenerate versions of quite a few special numbers and polynomials are introduced and investigated by means of various methods. The aim of this paper is to study some results on degenerate harmonic numbers, degenerate hyperharmonic numbers, degenerate Fubi polynomials and degenerate r-Fubini polynomials from a general identity which is valid for any two formal power series and involves the degenerate r-Stirling numbers of the second kind.
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.
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.
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.
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。
The aim of this paper is twofold. The first one is to find several relations between the type 2 higher-order degenerate Euler polynomials and the type 2 higher-order Changhee polynomials in connection with the degenerate stirling numbers of both kinds and Jindalrae-stirling numbers of both kinds. The second one is to introduce the discrete harmonic numbers and some related polynomials and numbers, and to derive their explicit expressions and an identity.
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.
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.
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.
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.
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.
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.
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.
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。
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.
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。
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.
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. …
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.
Harmonic Tutte polynomials of matroids II
本文提出有限Frobenius环上码的调和m重重量枚举器，建立调和MacWilliams型与Greene型恒等式，并给出自对偶码相对不变空间的结构。
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。
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.
The Waring Problem of Harmonic Polynomials
本文研究了调和多项式的瓦ring问题，证明其瓦ring秩等于次数，并提供了计算最小瓦ring分解的算法。
We show various supercongruences for truncated series which involve central binomial coefficients and harmonic numbers. The corresponding infinite series are also evaluated.
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。
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…
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…
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.
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.
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。
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.
This paper is now part of the new paper "Series with Hermite polynomials and applications" arXiv:1710.00687.
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