A complement to a recent paper on some infinite sums with the zeta values

A note on some infinite sums of Hurwitz zeta functions

We consider some closed-form evaluations of certain infinite sums involving the Hurwitz zeta function $ζ(s,α)$ of the form \[\sum_{k=1}^\infty (\pm 1)^k k^m ζ(s,k),\] where $m$ is a non-negative integer. For the sums with $m=0$ and the argument $k$ in $ζ(s,k)$ replaced by $ka+b$, where $a$ and $b$ are positive parameters, we also obtain a transformation formula suitable for computation in the limit $a\to0$.

Generating functions for multiple zeta star values

We study generating functions for multiple zeta star values in general form. These generating functions provide a connection between multiple zeta star values and multiple Euler sums, which allows us to express each multiple zeta star value in terms of multiple alternating Euler sums, and specifically, reduce the length of blocks of twos in the resulting sums.

Integrals of products of Hurwitz zeta functions via Feynman parametrization and two double sums of Riemann zeta functions

We consider two integrals over $x\in [0,1]$ involving products of the function $ζ_1(a,x)\equiv ζ(a,x)-x^{-a}$, where $ζ(a,x)$ is the Hurwitz zeta function, given by $$\int_0^1ζ_1(a,x)ζ_1(b,x)\,dx\quad\mbox{and}\quad \int_0^1ζ_1(a,x)ζ_1(b,1-x)\,dx$$ when $\Re (a,b)>1$. These integrals have been investigated recently in \cite{SCP}; here we provide an alternative derivation by application of Feynman parametrization. We also discuss a moment integral and the evaluation of two doubly infinite sums containing the Riemann zeta function $ζ(x)$ and two free parameters $a$ and $b$. The limiting forms of…

Bessel函数级数作为Riemann zeta函数级数的表示

Series over Bessel functions as series in terms of Riemann's zeta function

本文利用Hurwitz公式和三角级数求和公式，将Bessel函数级数转化为Riemann zeta函数级数，并求得有限和。

Central Binomial Sums, Multiple Clausen Values and Zeta Values

We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). The study of non-alternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experimental results involving polylogarithms in the golden ratio. In the non-alternating case, there is a strong connection to polylogarithms of the sixth root of unity, encountered in the 3-loop Feynman diagrams of {\tt hep-th/9803091} and subsequently in hep-ph/…

Euler Sums of Hyperharmonic Numbers

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: σ(r,m)=\sum_{n=1}^{\infty}((h_{n}^{(r)})/(n^{m})) can be expressed in terms of series of Hurwitz zeta function values. This is a generalization of a result of Mező and Dil. We also provide an explicit evaluation of σ(r,m) in a closed form in terms of zeta values and Stirling numbers of the first kind. Furthermore, we evaluate several other series involving hyperharmonic numbers.