Related papers: On generalized harmonic numbers, Tornheim double s…
By using the elementary symmetric polynomials and some results of number theory, we solve the well known problem of Lehmer on Euler's totient function. As application, we obtain a new characterization of prime numbers.
In recent three--loop calculations of massive Feynman integrals within Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the so-called generalized harmonic sums (in short $S$-sums) arise. They are characterized by…
It is conjectured that the regularized double shuffle relations give all algebraic relations among the multiple zeta values, and hence all other algebraic relations should be deduced from the regularized double shuffle relations. In this…
The Euler-Poincar\'e characteristic of a finite-dimensional Lie algebra vanishes. If we want to extend this result to Lie superalgebras, we should deal with infinite sums. We observe that a suitable method of summation, which goes back to…
We study analytic properties of multiple zeta-functions of generalized Hurwitz-Lerch type. First, as a special type of them, we consider multiple zeta-functions of generalized Euler-Zagier-Lerch type and investigate their analytic…
We prove that a Poisson-Newton formula, in a broad sense, is associated to each Dirichlet series with a meromorphic extension to the whole complex plane of finite order. These formulas simultaneously generalize the classical Poisson formula…
We unify the recently developed abstract theories of universal series and extended universal series to include sums of the form $\sum_{k=0}^n a_k x_{n,k}$ for given sequences of vectors $(x_{n,k})_{n\geq k\geq 0}$ in a topological vector…
We examine values of certain Tornheim's type of double series with odd weight. As a result, an affirmative answer to a conjecture about the parity theorem for the zeta function of the root system of the exceptional Lie algebra $G_2$,…
Multiple q-zeta values are a 1-parameter generalization (in fact, a q-analog) of the multiple harmonic sums commonly referred to as multiple zeta values. These latter are obtained from the multiple q-zeta values in the limit as q tends to…
In this paper, we derive eight basic identities of symmetry in three variables related to generalized Euler polynomials and alternating generalized power sums. All of these are new, since there have been results only about identities of…
Certain generalization of Euler numbers was defined in 1935 by Lehmer using cubic roots of unity, as a natural generalization of Bernoulli and Euler numbers. In this paper, Lehmer's generalized Euler numbers are studied to give certain…
In this paper we investigate congruence relationships of particular finite generalized harmonic numbers sums. We suggest more transparent and simpler method to analyse these sums and present several additional results for certain special…
In this paper, we show that the regularized determinants of some Dirichlet series are multiplicative. As an application, we give generalizations of Lerch's formula for the classical gamma function and we determine the sum of some Dirichlet…
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…
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…
In the framework of generalized Oppenheim expansions we prove strong law of large numbers for lightly trimmed sums. In the first part of this work we identify a particular class of expansions for which we provide a convergence result…
An arbitrary-depth reduction theorem for the `convolution' multiple L-values of Euler-Zagier type is proven by an analytic method. To this end, generalized polylogarithms associated to Dirichlet characters are defined. The proof uses the…
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…
In this paper, we will define general Eulerian numbers and Eulerian polynomials based on general arithmetic progressions. Under the new definitions, we have been successful in extending several well-known properties of traditional Eulerian…
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…