Related papers: The evaluation of Tornheim double sums. Part 1
We present another expression to regularize the Euler product representation of the Riemann zeta function. % in this paper. The expression itself is essentially same as the usual Euler product that is the infinite product, but we define a…
We prove two fast formulas for the Hurwitz values $\zeta(2,a)$ and $\zeta(3,a)$ respectively with the help of the WZ method. In them $(a)_n$ denotes the rising factorial or Pochhammer's symbol defined by $(a)_0=1$ and…
The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application…
The main purpose and motivation of this article is to create a linear transformation on the polynomial ring of rational numbers. A matrix representation of this linear transformation based on standard fundamentals will be given. For some…
Let Q(4n,d) be the sum of all multiple zeta values of depth d and weight 4n whose arguments are all multiples of 4. In this paper we derive a formula of Q(4n,d) for all d>2 as a finite sum involving binomial coefficients, Bernoulli numbers…
We derive several identities for the Hurwitz and Riemann zeta functions, the Gamma function, and Dirichlet $L$-functions. They involve a sequence of polynomials $\alpha_k(s)$ whose study was initiated in an earlier paper. The expansions…
We present expressions in terms of a double infinite series for the Stieltjes constants $\gamma_k(a)$. These constants appear in the regular part of the Laurent expansion for the Hurwitz zeta function. We show that the case…
Euler's sum formula and its multi-variable and weighted generalizations form a large class of the identities of multiple zeta values. In this paper we prove a family of identities involving Bernoulli numbers and apply them to obtain…
An easy generalization of Beukers' integrals allows us to conjecture a double integral formula involving the zeta and the gamma functions. A special case of this formula is Sondow's double integral formula for Euler's constant gamma.
An extension of two finite trigonometric series is studied to derive closed form formulae involving the Hurwitz-Lerch zeta function. The trigonometric series involves angles with a geometric series involving the powers of 3. These closed…
In this paper we shall define a special-valued multiple Hurwitz zeta functions, namely the multiple $t$-values $t(\boldsymbol{\alpha})$ and define similarly the multiple star $t$-values as $t^{\star}(\boldsymbol{\alpha})$. Then we consider…
We shall show that the sum of the series formed by the so-called hyperharmonic numbers can be expressed in terms of the Riemann zeta function. More exactly, we give summation formula for the general hyperharmonic series.
We study reciprocity formulas for Dedekind sums associated with absolutely continuous functions, extending the classical Dedekind-Rademacher reciprocity formula. In particular, we treat the case of periodic Bernoulli functions. Our approach…
We present explicit formulas for the following family of parametric binomial sums involving harmonic numbers for $p=0,1,2$ and $|t|\leq1$. $$ \sum_{k=1}^{\infty}\frac{H_{k-1}t^k}{k^p\binom{n+k}{k}}\quad \mbox{and}\quad…
The sum $S(h,k):=\sum_{j=1}^{k-1}(-1)^{j+1+[hj/k]}$ appears in the modular transformation formulae of the classical theta function $\vartheta_3(z)$. The double sum $S(k) := \sum_{h=1}^{k-1}S(h,k)$ has a remarkable distribution of values.…
We produce formulas for $$\sum_{j=1}^{2^{n-2}}\frac{1}{\sin^s\left(\frac{(2j-1)\pi}{2^n}\right)}$$ in terms of Generalized Bernoulli and Euler polynomials and use one of the formulas to produce a nice integral representation of the Riemann…
We introduce a new generalization of Stirling numbers of the second kind and analyze their properties, including generating functions, integral representations, and recurrence relations. These numbers are used to approximate Riemann zeta…
The hyperharmonic numbers h_{n}^{(r)} are defined by means of the classical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers: {\sigma}(r,m)=\sum_{n=1}^{\infty}((h_{n}^{(r)})/(n^{m})) can be expressed in terms of…
We establish closed-form expressions for the infinite series sum from n=2 to infinity of arctanh(n^-k) for all integers k >= 2 by connecting these sums to infinite product formulas involving the gamma function. Our approach uses logarithmic…
An alternative formula is presented for the evaluation of the zeta function values $\zeta(2k)$ without the need for Bernoulli numbers. Our formula is recursive, and improves the efficiency with which we can calculate large values of the…