Related papers: The evaluation of Tornheim double sums. Part 1

In this paper, we generalize the partial fraction decomposition which is fundamental in the theory of multiple zeta values, and prove a relation between Tornheim's double zeta functions of three complex variables. As applications, we give…

Using Euler transformation of series we relate values of Hurwitz zeta function at integer and rational values of arguments to certain rapidly converging series where some generalized harmonic numbers appear. The form of these generalized…

In this paper, we establish Kronecker limit type formulas for the generalized Mordell--Tornheim zeta function $\Theta(r,s,t,x)$ as a function of the third variable, in terms of Riemann-zeta and Gamma values. We also give series evaluations…

For the function $f(m,p,q,n)$, where $k,s,a$ general complex numbers and $q$ any positive integer, we establish the sum of values of the Hurwitz-Lerch zeta function $\Phi(f(m,p,q,n),k,a)$ taken at prime numbers $n$. Special cases of this…

This study presents explicit evaluations of the series \begin{equation*} \sum_{k=1}^\infty \frac{H_{k/n}^{(p)}}{k^q} \quad \text{and} \quad \sum_{k=1}^\infty \frac{(-1)^k H_{k/2n}^{(p)}}{k^q}, \quad p,q,n \in \mathbb{Z}_{\ge 1},\; q \ne 1,…

We introduce some new higher dimensional generalizations of the Dedekind sums associated with the Bernoulli functions and of those Hardy sums which are defined by the sawtooth function. We generalize a variant of Parseval's formula for the…

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,…

Using a different approach, we derive integral representations for the Riemann zeta function and its generalizations (the Hurwitz zeta, $\zeta(-k,b)$, the polylogarithm, $\mathrm{Li}_{-k}(e^m)$, and the Lerch transcendent,…

In the present paper, for $s,t,u \in {\mathbb{C}}$, we show rapidly (or globally) convergent series representations of the Tornheim double zeta function $T(s,t,u)$ and (desingularized) symmetric Tornheim double zeta functions. As a…

We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the…

The Watson-Harkins sum involving the product of the cosine and cosecant functions is extended to derive the finite sum of generalized Hurwitz-Lerch Zeta functions is derived in terms of the Hurwitz-Lerch Zeta function. A transformation…

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…

We consider certain double series of Eisenstein type involving hyperbolic-sine functions. We define certain generalized Hurwitz numbers, in terms of which we evaluate those double series. Our main results can be regarded as a certain…

We present a new proof of Euler's formulas for $\zeta(2k)$, where $k = 1,2,3,...$, which uses only the defining properties of the Bernoulli polynomials, obtaining the value of $\zeta(2k)$ by summing a telescoping series. Only basic…

We establish a connection between a function and a series representation using a similar technique with that that Euler used to solve the Basel problem. Our result concerns a more general series from which one can obtain $\zeta(2k)$ as a…

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…

In this work we derive and evaluate some infinite integrals involving the product of a generalized logarithm and polynomial functions in the denominator. These integrals are expressed in terms of finite series involving the Hurwitz-Lerch…

We develop series representations for the Hurwitz and Riemann zeta functions in terms of generalized Bernoulli numbers (N\"{o}rlund polynomials), that give the analytic continuation of these functions to the entire complex plane. Special…

A formula for the Hurwitz zeta function at the positive integers $k$, $\zeta(k,b)$, is created by solving the real and the imaginary parts separately and then combining them. A few different formulae for the Hurwitz zeta function are known…

The Stieltjes constants $\gamma_k(a)$ appear in the regular part of the Laurent expansion of the Hurwitz zeta function about its only polar singularity at $s=1$. We present multi-parameter summation relations for these constants that result…