Related papers: Some Evaluation of Quadratic Euler Sums
In this work we present the computer algebra package HarmonicSums and its theoretical background for the manipulation of harmonic sums and some related quantities as for example Euler-Zagier sums and harmonic polylogarithms. Harmonic sums…
We present explicit expressions for multi-fold logarithmic integrals that are equivalent to sums over polygamma functions at integer argument. Such relations find application in perturbative quantum field theory, quantum chemistry, analytic…
This paper provides a technique for evaluating some nonlinear Gaussian sums in closed forms. The evaluation is obtained from the known values of simpler exponential sums.
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…
We will employ the method of contour integration to investigate the parity results of non-embedded cyclotomic multiple $t$-values, which we refer to as cyclotomic Euler $T$-sums. We can provide explicit parity formulas for the linear and…
Many $\mathbb{Q}$-linear relations exist between multiple zeta values, the most interesting of which are various weighted sum formulas. In this paper, we generalized these to Euler sums and some other variants of multiple zeta values by…
We obtain the quadratic term in Euler's asymptotic expansion of the nth harmonic number by a simple modification of Young's elementary determination of the linear term.
The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin…
In this sequel to arXiv:0905.3327, we continue to study the congruence properties of the alternating version of multiple harmonic sums. As contrast to the study of multiple harmonic sums where Bernoulli numbers and Bernoulli polynomials…
By some hypergeometric summation theorems, the authors establish a series of new infinite summation formulas involving generalized harmonic numbers related to Riemann-Zeta function, with three different patterns.
We prove sum formulas for double polylogarithms of Hurwitz type, that is, involving a shifting parameter $b$ in the denominator. These formulas especially imply well-known sum formulas for double zeta values, and sum formulas for double…
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…
In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.
Summation formulas, such as the Euler-Maclaurin expansion or Gregory's quadrature, have found many applications in mathematics, ranging from accelerating series, to evaluating fractional sums and analyzing asymptotics, among others. We show…
In this paper we investigate the properties of the Euler functions. By using the Fourier transform for the Euler function, we derive the interesting formula related to the infinite series. Finally we give some interesting identities between…
Five series are evaluated in terms of zeta values. Three of the series involve harmonic numbers and one involves Stirling numbers of the first kind. The evaluation of these series is reduced to the evaluation of certain integrals, including…
This paper describes algorithms to deal with nested symbolic sums over combinations of harmonic series, binomial coefficients and denominators. In addition it treats Mellin transforms and the inverse Mellin transformation for functions that…
The harmonic polylogarithms (hpl's) are introduced. They are a generalization of Nielsen's polylogarithms, satisfying a product algebra (the product of two hpl's is in turn a combination of hpl's) and forming a set closed under the…
The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order.…
In this paper, we derive some interesting symmetric properties for the geenralized Euler numbers and polynomials.