Related papers: Euler sums of generalized 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…
Combining the derivative operator with a binomial sum from the telescoping method, we establish a family of summation formulas involving generalized harmonic numbers.
In this paper, we employ methods of contour integration and residue calculus to investigate the parity of two classes of cyclotomic Euler-type sums. One class involves products of cyclotomic harmonic numbers, while the other involves…
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…
In the present paper we generalize the Eulerian numbers (also of the second and third orders). The generalization is connected with an autonomous first-order differential equation, solutions of which are used to obtain integral…
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…
In this paper, we consider representations of integers as sums of generalized heptagonal numbers with a prescribed number of repeats of each heptagonal number appearing in the sum. In particular, we investigate the classification of such…
We consider different generalizations of the Euler formula and discuss the properties of the associated trigonometric functions. The problem is analyzed from different points of view and it is shown that it can be formulated in a natural…
In terms of the derivative operator and three hypergeometric series identities, several interesting summation formulas involving generalized harmonic numbers are established.
A generalization of the classical Lipschitz summation formula is proposed. It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of Bernoulli--type polynomials. Related families of…
Combining the derivative operator with Chu-Vandermonde convolution, we establish a class of summation formulas on generalized harmonic numbers.
In this paper we introduce the generalization of Multi Poly-Euler polynomials and we investigate some relationship involving Multi Poly-Euler polynomials. Obtaining a closed formula for generalization of Multi Poly-Euler numbers therefore…
Following an idea due to Euler, we evaluate the alternating sums of powers of consrcutive integers.
Translation from the Latin original, "Inventio summae cuiusque seriei ex dato termino generali" (1735). E47 in the Enestrom index. In this paper Euler derives the Euler-Maclaurin summation formula, by expressing y(x-1) with the Taylor…
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.…
In this paper we present a generalization of Faulhaber's formula to sums of arbitrary complex powers $m\in\mathbb{C}$. These summation formulas for sums of the form $\sum_{k=1}^{\lfloor x\rfloor}k^{m}$ and $\sum_{k=1}^{n}k^{m}$, where…
In terms of the derivative operator, integral operator and Saalsch\"{u}tz's theorem, two families of summation formulae involving generalized harmonic numbers are established.
In this paper, we consider three families of numerical series with general terms containing the harmonic numbers, and we use simple methods from classical and complex analysis to find explicit formulas for their respective sums.
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…
Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices. It is shown that there are other linear combinations of the numbers of simplices which are combinatorial…