Search

We determine the values of several infinite series involving the Riemann zeta function. In particular, degree one rationally weighted summands involving the Riemann function evaluated at even numbers give a finite sum involving only the Riemann function evaluated at odd numbers. [ABSTRACT FROM AUTHOR]

This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers Hn(p,q) ... 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 the Riemann zeta values. [ABSTRACT FROM AUTHOR]

A very recent article delved into and expanded the four parametric linear Euler sums, revealing that two well-established subjects—Euler sums and series involving the zeta functions—display particular correlations. In this study, we present several closed forms of series involving zeta functions by using formulas for series associated with the zeta functions detailed in the aforementioned paper. Another closed form of series involving Riemann zeta functions is provided by utilizing a known identity for a series of rational functions in the series index, expressed in terms of Gamma functions. Furthermore, we demonstrate a myriad of applications and relationships of series involving the zeta functions and the extended parametric linear Euler sums. These include connections with Wallis's infinite product formula for π , Mathieu series, Mellin transforms, determinants of Laplacians, certain integrals expressed in terms of Euler sums, representations and evaluations of some integrals, and certain parametric Euler sum identities. The use of Mathematica for various approximation values and certain integral formulas is elaborated upon. Symmetry naturally occurs in Euler sums. [ABSTRACT FROM AUTHOR]

In this paper, we evaluate in closed form several different series involving the harmonic numbers and skew-harmonic numbers. We consider two classes of series involving these sequences. One class of series involves the product of the nth harmonic or skew-harmonic number and a tail. We provide the solution to two open problems concerning these harmonic series with tails from Alina SÎntămărian and Ovidiu Furdui's book Sharpening Mathematical Analysis Skills. The other class of series is the Hardy series, which involves a logarithm and the Euler-Mascheroni constant being subtracted from the nth harmonic number. [ABSTRACT FROM AUTHOR]

A very recent article delved into and expanded the four parametric linear Euler sums, revealing that two well-established subjects—Euler sums and series involving the zeta functions—display particular correlations. In this study, we present several closed forms of series involving zeta functions by using formulas for series associated with the zeta functions detailed in the aforementioned paper. Another closed form of series involving Riemann zeta functions is provided by utilizing a known identity for a series of rational functions in the series index, expressed in terms of Gamma functions. Furthermore, we demonstrate a myriad of applications and relationships of series involving the zeta functions and the extended parametric linear Euler sums. These include connections with Wallis’s infinite product formula for π, Mathieu series, Mellin transforms, determinants of Laplacians, certain integrals expressed in terms of Euler sums, representations and evaluations of some integrals, and certain parametric Euler sum identities. The use of Mathematica for various approximation values and certain integral formulas is elaborated upon. Symmetry naturally occurs in Euler sums.

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function η(s), and hence Riemann's function ζ(s), is obtained in terms of the Exponential Integral function Es(iκ) of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are unearthed. The incomplete functions ζ±(s) and η±(s) are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths between the endpoints. [ABSTRACT FROM AUTHOR]

In this paper, the authors investigate two special families of series involving the reciprocal central binomial coefficients and Lucas numbers. Connections with several familiar sums representing the integer-valued Riemann zeta function are also pointed out.

In this paper, by the Faà di Bruno formula, we establish the decompositions of two general fractions involving the reciprocals of products of binomial coefficients. Using the decompositions, we discuss the evaluations of some Euler-type sums involving harmonic numbers and binomial coefficients, such as and some other forms. We present some explicit evaluations as examples and provide the Maple package to compute the sums and. It can be found that this work gives a unified approach to such sums and generalizes many known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we extend tools developed in [26] to study harmonic numbers sums. We use these objects to compute Euler-related sums and to explore their connections with classical Euler sums. We establish some new relations between Euler sums and related sums. The relations obtained allow us to find some nice closed-form representations of nonlinear Euler sums through zeta values and linear sums. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the integral of x n log p (sin (x)) for natural numbers n and p. In doing so, we recover some well-known results and remark on some relations to the log-sine integral Ls n + p + 1 (n) (θ). Later, we use properties of Bell polynomials to find an expression for the derivative of the central binomial and shifted central binomial coefficients as finite sums of polygamma functions and harmonic numbers. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For any we first give new proofs for the following well-known combinatorial identities and and then we produce the generating function and an integral representation for . Using them we evaluate many interesting finite and infinite harmonic sums in closed form. For example, we show that and where are generalized harmonic numbers defined below. [ABSTRACT FROM AUTHOR]

In this paper, we develop an approach to evaluation of nonlinear Euler sums. The approach is based on Tornheim type series computations. By the approach, we can obtain some closed form representations of quadratic and cubic sums in terms of zeta values and linear sums. Furthermore, we also evaluate several other series involving harmonic numbers. Some interesting new consequences and illustrative examples are considered. [ABSTRACT FROM AUTHOR]

This paper develops an approach to evaluation of Euler related sums. The approach is based on simple integral computations. By the approach, we can obtain some closed form representations of sums of quadratic and cubic harmonic numbers and reciprocal binomial coefficients. The given representations are new. [ABSTRACT FROM AUTHOR]

The paper is about calculating multiple fractional part integrals of the form ...where {x} denotes the fractional part of x and k,m and p are nonnegative integers. We show that these integrals can be expressed as series involving products of Riemann zeta function values and some binomial coefficients. We obtain, as particular cases of our results, new integral representations of Euler's constant as double and triple symmetric fractional part integrals. [ABSTRACT FROM AUTHOR]

In this paper, we first develop a set of identities for Euler-type sums. We then investigate products of the shifted harmonic numbers and the reciprocal binomial coefficients. We briefly indicate relevant connections of the results presented here with those given by earlier authors. As by-products of our investigation, we derive several (presumably new) one-parameter and two-parameter summation formulas for the hypergeometric series $$\;_3F_2\;$$ at argument 1. [ABSTRACT FROM AUTHOR]

The paper is about calculating the following class of quadratic series:wherek≥2 is an integer. We show thatSkcan be expressed as a rational linear combination of zeta function values, binomial coefficients and products of two zeta function values. [ABSTRACT FROM AUTHOR]

The multiple gamma function Γn(z), defined by a recurrence-functional equation as a generalization of the Euler gamma function, is used in many applications of pure and applied mathematics, and theoretical physics. The theory of the multiple gamma function has been related to certain spectral functions in mathematical physics, to the study of functional determinants of Laplacians of then-sphere, to HeckeL-functions, to the Selberg zeta function, and to the random matrix theory. There is a wide class of definite integrals and infinite sums appearing in statistical physics (the Potts model) and the lattice theory, which can be computed by means of the Γn(z) function. This paper presents new integral representations for the multiple gamma function and other mathematical functions and constants. [ABSTRACT FROM AUTHOR]

This paper presents a family of new integral representations and asymptotic series of the multiple gamma function. The numerical schemes for high-precision computation of the Barnes gamma function and Glaisher's constant are also discussed. [ABSTRACT FROM AUTHOR]

From the Kummer 2 F 1-summation theorem, the Dixon–Kummer 4 F 3-summation theorem and the Dougall–Dixon 5 F 4-summation theorem, we establish, by means of the Bell polynomials, three general formulas related to the generalized harmonic numbers and the Riemann zeta function. Based on these three general formulas, we further find series of harmonic number identities. Some of these identities involve both finite summation and infinite series, so that we can determinate the explicit expressions of numerous infinite series. In particular, we show that several interesting analogues of the Euler sums can be evaluated. [ABSTRACT FROM AUTHOR]

Abstract: Harmonic numbers and generalized harmonic numbers have been studied since the distant past and are involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we aim at presenting further interesting identities about certain finite or infinite series involving harmonic numbers and generalized harmonic numbers. We also give a brief eclectic review on some known properties associated with the harmonic and generalized harmonic numbers. [Copyright &y& Elsevier]

We extend some results of Euler related sums. Integral and closed-form representations of sums with products of harmonic numbers and binomial coefficients are developed in terms of Polygamma functions. The various representations presented in this paper are believed to be new. [ABSTRACT FROM AUTHOR]

Since the time of Euler, the so-called Euler-Mascheroni constant γ has been involved in a wide variety of important mathematical formulas and results. Among these formulas and results, a considerably large number of integral representations of γ have been developed. Here we aim at investigating several further integral expressions for the Euler-Mascheroni constant γ. [ABSTRACT FROM AUTHOR]

Abstract: We consider summations over digamma and polygamma functions, often with summands of the form and , where m, p, q, and r are positive integers. We develop novel general integral representations and present explicit examples. Special cases of the sums reduce to known linear Euler sums. The sums of interest find application in quantum field theory, including evaluation of Feynman amplitudes. [Copyright &y& Elsevier]

Abstract: The history of problems of evaluation of series associated with the Riemann Zeta function can be traced back to Christian Goldbach (1690–1764) and Leonhard Euler (1707–1783). Many different techniques to evaluate various series involving the Zeta and related functions have since then been developed. The authors show how elegantly certain families of series involving the Hurwitz–Lerch Zeta function can be evaluated by starting with a single known identity for the Hurwitz–Lerch Zeta function. Some of the special cases of these series identities are also shown to lead to certain known families of summation formulas involving the Hurwitz Zeta function. [Copyright &y& Elsevier]

A fascinatingly large number of seemingly independent solutions of the so-called Basler problem of evaluating the Riemann Zeta function ζ(s) when s=2, which was of vital importance to Leonhard Euler (1707–1783) and the Bernoulli brothers (Jakob Bernoulli (1654–1705) and Johann Bernoulli (1667–1748)), have appeared in the mathematical literature ever since Euler first solved this problem in the year 1736. The main object of this two-part series of lectures is to present some recent developments on the evaluations and representations of ζ(s) when s∈N&z.drule;{1}, N being the set of natural numbers. We emphasize upon several interesting classes of rapidly convergent series representations for ζ(2n+1) (n∈N) which have been developed in recent years. In two of many computationally useful special cases considered here, it is observed that ζ(3) can be represented by means of series which converge much more rapidly than that in Euler’s celebrated formula as well as the series used recently by Roger Ape´ry (1916–1994) in his proof of the irrationality of ζ(3). Symbolic and numerical computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of one of these series are capable of producing an accuracy of seven decimal places. [Copyright &y& Elsevier]

In this paper, the authors investigate two special families of series involving the reciprocal central binomial coefficients and Lucas numbers. Connections with several familiar sums representing the integer-valued Riemann zeta function are also pointed out. [ABSTRACT FROM AUTHOR]

The core of the present paper is represented by the calculation of two essential harmonic series with a weight 5 structure, involving harmonic numbers of the type H2n. The two main series are evaluated by also exploiting results and strategies presented in the book, (Almost) Impossible Integrals, Sums, and Series, 2019. [ABSTRACT FROM AUTHOR]