In this short paper, we derive an integral representation for Euler sums of hyperharmonic numbers. We use results established by other authors to then show that the integral has a closed-form in terms of zeta values and Stirling numbers of the first kind. Specifically, the integral has the form of $$\int_0^\infty \frac{t^{m-1}\ln(1-e^{-t})}{(1-e^{-t})^r} \ dt$$ where $m, r \in \mathbb{N}$, $m > r$ and $r\ge1$.
In this paper, we mainly show that Euler sums of generalized hyperharmonic numbers can be expressed in terms of linear combinations of the classical Euler sums.
In this paper, we mainly show that generalized hyperharmonic number sums with reciprocal binomial coefficients can be expressed in terms of classical (alternating) Euler sums, zeta values and generalized (alternating) harmonic numbers.
In this paper, we introduce a new type of generalized alternating hyperharmonic numbers $H_n^{(p,r,s_{1},s_{2})}$, and show that Euler sums of the generalized alternating hyperharmonic numbers $H_n^{(p,r,s_{1},s_{2})}$ can be expressed in terms of linear combinations of classical (alternating) Euler sums.
In this paper, we derive a formula for the sums of powers of the first $n$ positive integers, $S_k(n)$, that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then, using an explicit representation for the hyperharmonic numbers, we generalize this formula to the sums of powers of an arbitrary arithmetic progression. Moreover, we express the Bernoulli polynomials in terms of hyperharmonic polynomials and Stirling numbers of the second kind. Finally, we extend the obtained formula for $S_k(n)$ to negative values of $n$.
The generalized hyperharmonic numbers $h_n^{(m)}(k)$ are defined by means of the multiple harmonic numbers. We show that the hyperharmonic numbers $h_n^{(m)}(k)$ satisfy certain recurrence relation which allow us to write them in terms of classical harmonic numbers. Moreover, we prove that the Euler-type sums with hyperharmonic numbers: \[S\left( {k,m;p} \right): = \sum\limits_{n = 1}^\infty {\frac{{h_n^{\left( m \right)}\left( k \right)}}{n^p}} \;\;\left(p\geq m+1,\ {k = 1,2,3} \right)\] can be expressed as a rational linear combination of products of Riemann zeta values and harmonic numbers.…
A number of identities are proved by using Stirling transforms. These identities involve Stirling numbers of the first and second kinds, hyperharmonic and derangement numbers, Bernoulli and Euler numbers and polynomials, powers, power sums, and factorials.
Recently, Dil and Boyadzhiev \cite{AD2015} proved an explicit formula for the sum of multiple harmonic numbers whose indices are the sequence $\left( {{\left\{ 0 \right\}_r},1} \right)$. In this paper we show that the sums of multiple harmonic numbers whose indices are the sequence $\left( {{\left\{ 0 \right\}_r,1};{\left\{ 1 \right\}_{k-1}}} \right)$ can be expressed in terms of (multiple) zeta values, multiple harmonic numbers and Stirling numbers of the first kind, and give an explicit formula.
In this paper we consider the Daehee numbers and polynomials of the first and second kind, and give several explicit representations for them. In particular, we express the Daehee polynomials as the derivative of a generalized binomial coefficient. This is done by performing the Stirling transform of the power sum polynomial $S_k(x)$ associated with the sum of $k$th powers of the first $n$ positive integers $S_k(n) = 1^k + 2^k + \cdots + n^k$. Furthermore, we show the relationship between the Daehee polynomials and the hyperharmonic polynomials. This allows us to also express the hyperharmonic…
We give explicit evaluations of the linear and non-linear Euler sums of hyperharmonic numbers $h_{n}^{\left( r\right) }$ with reciprocal binomial coefficients. These evaluations enable us to extend closed form formula of Euler sums of hyperharmonic numbers to an arbitrary integer $r$. Moreover, we reach at explicit formulas for the shifted Euler-type sums of harmonic and hyperharmonic numbers. All the evaluations are provided in terms of the Riemann zeta values, harmonic numbers and linear Euler sums.
The hyperharmonic numbers h_{n}^{(r)} are defined by means of the classical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers: σ(r,m)=\sum_{n=1}^{\infty}((h_{n}^{(r)})/(n^{m})) can be expressed in terms of series of Hurwitz zeta function values. This is a generalization of a result of Mező and Dil. We also provide an explicit evaluation of σ(r,m) in a closed form in terms of zeta values and Stirling numbers of the first kind. Furthermore, we evaluate several other series involving hyperharmonic numbers.
Linear harmonic number sums had been studied by a variety of authors during the last centuries, but only few results are known about nonlinear Euler sums of quadratic or even higher degree. The first systematic study on nonlinear Euler sums consisting of products of hyperharmonic sums had been published by Flajolet and Salvy in 1997 followed by similar studies presented during the last years by different authors. Although these studies had been restricted to sums where the nominator consists of a product of even or odd hyperharmonic sums, where the denominator is of the type $1/k^n$. We have g…
In this paper, we study the theory of the harmonic and the hyperharmonic Fibonacci numbers. Also, we get some combinatoric identities like as harmonic and hyperharmonic numbers and we obtain some useful formulas for $\mathbb{F}_{n}$, which is finite sums of reciprocals of Fibonacci numbers. We obtain spectral and Euclidean norms of circulant matrices involving harmonic and hyperharmonic Fibonacci numbers.
In this paper, we present two new generalizations of the Euler-Mascheroni constant arising from the Dirichlet series associated to the hyperharmonic numbers. We also give some inequalities related to upper and lower estimates, and evaluation formulas.
Recently, Kargin et al. (arXiv:2008.00284 [math.NT]) obtained (among many other things) the following formula for the hyper-sums of powers of integers $S_k^{(m)}(n)$ \begin{equation*} S_k^{(m)}(n) = \frac{1}{m!} \sum_{i=0}^{m} (-1)^i \genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1} S_{k+i}(n), \end{equation*} where $S_k^{(0)}(n) \equiv S_k(n)$ is the ordinary power sum $1^k + 2^k + \cdots + n^k$. In this note we point out that a formula equivalent to the preceding one was already established in a different form, namely, a form in which $\genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1}$ is given explicitly a…
This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers $H_{n}^{\left( p,q\right) }$ \[ ζ_{H^{\left( p,q\right) }}\left( r\right) =\sum\limits_{n=1}^{\infty }\dfrac{H_{n}^{\left( p,q\right) }}{n^{r}}% \] in terms of the famous Euler sums of generalized harmonic numbers. Moreover, several infinite series, whose terms consist of certain harmonic numbers and reciprocal binomial coefficients, are evaluated in terms of Riemann zeta values.
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.
In this paper, we study the alternating Euler $T$-sums and related sums by using the method of contour integration. We establish the explicit formulas for all linear and quadratic Euler $T$-sums and related sums. Some interesting new consequences and illustrative examples are considered.
The explicit formulas expressing harmonic sums via alternating Euler sums (colored multiple zeta values) are given, and some explicit evaluations are given as applications.
We give a complete and elementary proofs of "Jordan's sums" and study Euler's types sums. In particular we give a formula for the sum of series with same weight, which is similar to this one of classical 2-Euler's sums.
In this paper, we obtain some formulas for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. By using these formulas, we give new closed form sums of several quadratic Euler series through Riemann zeta values, polylogarithm functions and linear sums. Furthermore, some relationships between Euler sums and integrals of polylogarithm functions are established.
We define a parametric variant of generalized Euler sums and construct contour integration to give some explicit evaluations of these parametric Euler sums. In particular, we establish several explicit formulas of (Hurwitz) zeta functions, linear and quadratic parametric Euler sums. Furthermore, we also give an explicit evaluation of alternating double zeta values $\ze(\overline{2j},2m+1)$ in terms of a combination of alternating Riemann zeta values by using the parametric Euler sums.
In this paper we present a new family of identities for Euler sums and integrals of polylogarithms by using the methods of generating function and integral representations of series. Then we apply it to obtain the closed forms of all quadratic Euler sums of weight equal to ten. Furthermore, we also establish some relations between multiple zeta (star) values and nonlinear Euler sums. As applications of these relations, we give new closed form representations of several cubic Euler sums through single zeta values and linear sums. Finally, with the help of numerical computations of Mathematica o…
This paper develops an approach to the evaluation of quadratic Euler sums that involve harmonic numbers. The approach is based on simple integral computations of polyloga- rithms. By using the approach, we establish some relations between quadratic Euler sums and linear sums. Furthermore, we obtain some closed form representations of quadratic sums in terms of zeta values and linear sums. The given representations are new.
In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic Euler sums through Riemann zeta function and linear sums. The given representations are new.
In this paper we define the generalized q-analogues of Euler sums and present a new family of identities for q-analogues of Euler sums by using the method of Jackson q-integral rep- resentations of series. We then apply it to obtain a family of identities relating quadratic Euler sums to linear sums and q-polylogarithms. Furthermore, we also use certain stuffle products to evaluate several q-series with q-harmonic numbers. Some interesting new results and illustrative examples are considered. Finally, we can obtain some explicit relations for the classical Euler sums when q approaches to 1.
In this paper, we define a parametric variant of generalized Euler sums and call them the (alternating) parametric Euler $T$-sums. By using the contour integration method and residue theorem, we establish several explicit formulae for the linear parametric Euler $T$-sums. Furthermore, by applying the results, we obtain explicit formulae for the Hoffman's (alternating) double $t$-values and Kaneko-Tsumura's (alternating) double $T$-values.
The alternating multiple harmonic sums are partial sums of the infinite series defining the Euler sums which are the alternating version of the multiple zeta value series. In this paper, we present some systematic structural results of the van Hamme type congruences of these sums, collected as finite Euler sums. Moreover, we relate this to the structure of the Euler sums, which generalizes the corresponding result of the multiple zeta values. We also provide a few conjectures with extensive numerical support.
In this paper we are interested in Euler-type sums with products of harmonic numbers, Stirling numbers and Bell numbers. We discuss the analytic representations of Euler sums through values of polylogarithm function and Riemann zeta function. Moreover, we provide explicit evaluations for almost all Euler sums with weight 5, which can be expressed in terms of zeta values and polylogarithms. Furthermore, we give explicit formula for several classes of Euler-related sums in terms of zeta values and harmonic numbers, and several examples are given. The given representations are new.
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 study several variants of Euler sums by using the methods of contour integration and residue theorem. These variants exhibit nice properties such as closed forms, reduction, etc., like classical Euler sums. In addition, we also define a variant of multiple zeta values of level 2, and give some identities on relations between these variants of Euler sums and the variant of multiple zeta values.
For positive integers $p_1,p_2,\ldots,p_k,q$ with $q>1$, we define the Euler $T$-sum $T_{p_1p_2\cdots p_k,q}$ as the sum of those terms of the usual infinite series for the classical Euler sum $S_{p_1p_2\cdots p_k,q}$ with odd denominators. Like the Euler sums, the Euler $T$-sums can be evaluated according to the Contour integral and residue theorem. Using this fact, we obtain explicit formulas for Euler $T$-sums with repeated arguments analogous to those known for Euler sums. Euler $T$-sums can be written as rational linear combinations of the Hoffman $t$-values. Using known results for Hoffm…
it is the purpose of this paper to construct a p-adic continuous function for an odd prime to contain a p-adic q-analogue of higher order Dedekind type sums related to q-Euler polynomials and numbers.
We define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results of Flajolet and Salvy's paper \cite{FS1998} are the immediate corollaries of main results in this paper. Furthermore, we provide some parameterized extensions of Ramanujan-type identities that involve hyperbolic series. Some interesting new consequences and illustrative examples are considered.
We present several sequences of Euler sums involving odd harmonic numbers. The calculational technique is based on proper two-valued integer functions, which allow to compute these sequences explicitly in terms of zeta values only.
In this paper, we study the alternating Euler $T$-sums and $§$-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit formulas for the (alternating) linear and quadratic Euler $T$-sums and $§$-sums, from which, the parity theorems of Hoffman's double and triple $t$-values and Kaneko-Tsumura's double and triple $T$-values are further obtained. As supplements, we also show that the linear $T$-sums and $§$-sums are expressible in…
The classical Dedekind sums appear in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. The Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations, and are shown to satisfy some reciprocity relations. In contrast, Dedekind type DC (Daehee and Changhee) sums and their generalizations are defined in terms of Euler functions and their generalizations. The purpose of this paper is to introduce the poly-Dedekind type DC sums, which are obtained from the Dedekind type DC sums by…
In this paper we investigate some interesting of the (h,q)-extension of Euler numbers and polynomials. Finally, we will give some relations between these numbers anf polynomials
Flajolet and Salvy pointed out that every Euler sum is a $\mathbb{Q}$-linear combination of multiple zeta values. However, in the literature, there is no formula completely revealing this relation. In this paper, using permutations and compositions, we establish two explicit formulas for the Euler sums, and show that all the Euler sums are indeed expressible in terms of MZVs. Moreover, we apply this method to the alternating Euler sums, and show that all the alternating Euler sums are reducible to alternating MZVs. Some famous results, such as the Euler theorem, the Borwein--Borwein--Girgensoh…
In this paper, by using the method of Contour Integral Representations and the Theorem of Residues and integral representations of series, we discuss the analytic representa- tions of parametric Euler sums that involve harmonic numbers through zeta values and rational function series, either linearly or nonlinearly. Furthermore, we give explicit formulae for several parametric quadratic and cubic sums in terms of zeta values and rational series. Moreover, some interesting new consequences and illustrative examples are considered.
We establish some identities of Euler related sums. By using these identities, we discuss the closed form representations of sums of harmonic numbers and reciprocal parametric binomial coefficients through parametric harmonic numbers, shifted harmonic numbers and Riemann zeta function with positive integer arguments. In particular we investigate products of quadratic and cubic harmonic numbers and reciprocal parametric binomial coefficients. Some illustrative special cases as well as immediate consequences of the main results are also considered.
We construct an analytic approach to evaluate odd Euler sums, multiple zeta value $ζ(3,2,\ldots,2)$ and multiple $t$-value $t\left(3,2,\ldots,2\right)$. Moreover, we also conjecture a closed expression for multiple $t$-value $t\left(2,\ldots,2,1\right)$.
In the present paper, our goal is to introduce a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order dedekind-type sums with weight alpha related to Extended q-Euler polynomials by using p-adic q-integral in the p-adic integer ring.
We show that Euler's relation and the Taxi-Cab relation are both solutions of the same equation. General solutions of sums of two consecutive cubes equaling the sum of two other cubes are calculated. There is an infinite number of relations to be found among the sums of two consecutive cubes and the sum of two other cubes, in the form of two families. Their recursive and parametric equations are calculated.
In this note, we provide refined estimates of the following sums involving the Euler totient function: $$\sum_{n\le x} ϕ\left(\left[\frac{x}{n}\right]\right) \qquad \text{and} \qquad \sum_{n\le x} \frac{ϕ([x/n])}{[x/n]}$$ where $[x]$ denotes the integral part of real $x$. The above summations were recently considered by Bordellès et al. and Wu.
For finite sums of non-negative powers of arithmetic progressions the generating functions (ordinary and exponential ones) for given powers are computed. This leads to a two parameter generalization of Stirling and Eulerian numbers. A direct generalization of Bernoulli numbers and their polynomials follows. On the way to find the Faulhaber formula for these sums of powers in terms of generalized Bernoulli polynomials one is led to a one parameter generalization of Bernoulli numbers and their polynomials. Generalized Lah numbers are also considered.
We aim to investigate the four types of variant Euler harmonic sums. Also, as corollaries, we provide particular examples of our core findings, some of whose further instances are evaluated in terms of basic and well-known functions as well as certain mathematical constants. We explore relevant linkages between our results and those of other previously established studies. An examination of a specific case of one result shows a relationship to series involving zeta functions, which is also a popular area of research.
P. Flajolet and B. Salvy \cite{FS1998} prove the famous theorem that a nonlinear Euler sum $S_{i_1i_2\cdots i_r,q}$ reduces to a combination of sums of lower orders whenever the weight $i_1+i_2+\cdots+i_r+q$ and the order $r$ are of the same parity. In this article, we develop an approach to evaluate the cubic sums $S_{1^2m,p}$ and $S_{1l_1l_2,l_3}$. By using the approach, we establish some relations involving cubic, quadratic and linear Euler sums. Specially, we prove the cubic sums $S_{1^2m,m}$ and $S_{1(2l+1)^2,2l+1}$ are reducible to zeta values, quadratic and linear sums. Moreover, we pro…
We study three special Dirichlet series, two of them alternating, related to the Riemann zeta function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at poles). These values are given in terms of Bernoulli and Euler numbers.
We examine the sums $S(k,\,n)$ of the $k-$th powers of the $ϕ(n)$ integers $α_1<α_2<\cdots<α_{ϕ(n)}$ less than and prime to $n$ (Euler set) and prove a formula (new) for $S(3,\,n)$. If $n$ equals a prime $p$, we prove a theorem showing a connection of $S(3,\,p)$ with Artin's conjectural constant for primitive roots and with other functions involving primes.
In this paper, we investigate the Euler sums $$ G_{n+2}(p,q)=\sum_{1\leq k_1<k_2<\cdots<k_{p+1}}\frac1{k_1k_2\cdots k_pk_{p+1}^{n+2}} \sum_{1\leq\ell_1\leq\ell_2\leq\cdots\leq\ell_q\leq k_{p+1}}\frac1{\ell_1\ell_2\cdots\ell_q}. $$ We give another two representations, a reflection formula, and some other properties. Then we use these results to calculate $ζ^\star(r+2,\{2\}^n)$, for $r=0,1,2$, as our applications.
