Related papers: Central Binomial Sums, Multiple Clausen Values and…
We investigate a class of combinatorial sums involving reciprocals of central binomial coefficients , employing generating functions as the primary solution technique to formulate and analyze series involving the Catalan's constant. Using a…
We give a proof of double shuffle relations for $p$-adic multiple zeta values by developing higher dimensional version of tangential base points and discussing a relationship with two (and one) variable $p$-adic multiple polylogarithms.
We analyse a collection of mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae. Such examinations are performed both unconditionally and under the assumption of a weaker version of the $abc$…
We survey various results and conjectures concerning multiple polylogarithms and the multiple zeta function. Among the results, we announce our resolution of several conjectures on multiple zeta values. We also provide a new integral…
We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested in…
Functions satisfying the functional equation \begin{align*} \sum_{r=0}^{n-1} (-1)^r f(x+ry, ny) = f(x,y), \quad \text{for any positive odd integer $n$}, \end{align*} are named the alternating invariant functions. Examples of such functions…
We show that each member of a doubly infinite sequence of highly nonlinear expressions of Bernoulli polynomials, which can be seen as linear combinations of certain higher-order convolutions, is a multiple of a specific product of linear…
Recently, several new results related to the evaluation of the series sum (-1)^n zeta(n)/(n+k) were published. In this short note we show that this series also possesses an interesting connection to the values of the zeta-function on the…
An explicit identity of sums of powers of complex functions presented via this a closed-form formula of Riemann zeta function produced at any given non-zero complex numbers. The closed-form formula showed us Riemann zeta function has no…
The aim of this paper is to derive a summation formula for the alternating infinite series and an expression for zeta function by using hyperbolic secant random variables. These identities involve Euler numbers and are obtained by computing…
We study sums of the form $\sum_{k=m}^n a_{nk} b_{km}$, where $a_{nk}$ and $b_{km}$ are binomial coefficients or unsigned Stirling numbers. In a few cases they can be written in closed form. Failing that, the sums still share many common…
Commutative complex numbers of the form u=x+\alpha y+\beta z+\gamma t in 4 dimensions are studied, the variables x, y, z and t being real numbers. Four distinct types of multiplication rules for the complex bases \alpha, \beta and \gamma…
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…
A new class of alternating convolutions concerning binomial coefficients and Catalan numbers are evaluated in closed forms.
In recent years, there has been intensive research on the ${\mathbb Q}$-linear relations between multiple zeta (star) values. In this paper, we prove many families of identities involving the $q$-analog of these values, from which we can…
In this note, using an idea from \cite{Amo-Carrillo-Sanchez} we derive some new series representations involving $\zeta(2n)$ and Euler numbers. Using a well-known series representation for the Clausen function, we also provide some new…
We confirm several conjectures of Guo, Jouhet and Zeng concerning the factors of alternative binomials sums.
Recently, R. Tauraso established finite $p$-analogues of famous Ap\'ery series for $\zeta(2)$ and $\zeta(3).$ In this paper, we present several congruences for finite central binomial sums arising from the truncation of Ap\'ery-type series…
We define discrete nested sums over integer points for symbols on the real line, which obey stuffle relations whenever they converge. They relate to Chen integrals of symbols via the Euler-MacLaurin formula. Using a suitable holomorphic…
In this paper we prove some new identities for multiple zeta values and multiple zeta star values of arbitrary depth by using the methods of integral computations of logarithm function and iterated integral representations of series. By…