Related papers: Algebraic identities among $q$- analogue of Euler …
We provide a $q$-analogue of Euler's formula for $\zeta(2k)$ for $k\in\mathbb{Z}^+$. Our main results are stated in Theorems 3.1 and 3.2 below. The result generalizes a recent result of Z.W. Sun who obtained $q$-analogues of…
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…
Various new identities, recurrence relations, integral representations, connection and explicit formulas are established for the Bernoulli, Euler numbers and the values of Riemann's zeta function. To do this, we explore properties of some…
The aim of this paper is to present a general algebraic identity. Applying this identity, we provide several formulas involving the q-binomial coefficients and the q-harmonic numbers. We also recover some known identities including an…
In this paper, we use two different approaches to introduce $q$-analogs of Riemann's zeta function and prove that their values at even integers are related to the $q$-Bernoulli and $q$ Euler's numbers introduced by Ismail and Mansour…
An elementary method of computing the values at negative integers of the Riemann zeta function is presented. The principal ingredient is a new q-analogue of the Riemann zeta function. We show that for any argument other than 1 the classical…
In this paper, we give a purely algebraic proof of an identity coming directly from Euler's reflection formula for the gamma function. Our proof uses Hoffman's harmonic algebra and some binomial identities.
Recently, $\lambda$-Bernoulli and $\lambda$-Euler numbers are studied in [5, 10]. The purpose of this paper is to present a systematic study of some families of the $q$-extensions of the $\lambda$-Bernoulli and the $\lambda$-Euler numbers…
The double zeta function was first studied by Euler in response to a letter from Goldbach in 1742. One of Euler's results for this function is a decomposition formula, which expresses the product of two values of the Riemann zeta function…
In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative…
A q-analogue of the Riemann zeta function was studied in [Kaneko et al. 03] via a certain q-series of two variables. We introduce in a similar way a q-analogue of the Dirichlet L-functions and make a detailed study of them, including some…
Recently (see [1]) I has introduced an interesting the Euler-Barnes multiple zeta function. In this paper we construct the q-analogue of Euler-Barnes multiple zeta function which interpolates the q-analogue of Frobenius-Euler numbers of…
In this survey article, we discuss the algebraic structure of q-analogues of multiple zeta values, which are closely related to derivatives of Eisenstein series. Moreover, we introduce the formal double Eisenstein space, which generalizes…
Recently, the higher-order q-Euler polynomials and multiple q-Euler zeta functions are introduced by T. Kim ([8, 9]). In this paper, we investigate some symmetric properties of the multiple q-Euler zeta function and derive various…
We discuss some aspects of the search for identities using computer algebra and symbolic methods. The focus is on so-called Apery-like formulae for special values of the Riemann Zeta function. Much work lays ahead in formally proving and…
Following Bachmann's recent work on bi-brackets and multiple Eisenstein series, Zudilin introduced the notion of multiple q-zeta brackets, which provides a q-analog of multiple zeta values possessing both shuffle as well as quasi-shuffle…
In this paper, we derive eight basic identities of symmetry in three variables related to $q$-Euler polynomials and the $q$-analogue of alternating power sums. These and most of their corollaries are new, since there have been results only…
Some identities for the Riemann zeta-function are proved, using properties of the Mellin transform and M\"untz's identity.
Nous \'etudions la nature arithm\'etique de $q$-analogues des valeurs $\zeta(s)$ de la fonction z\^eta de Riemann, notamment des valeurs des fonctions $\zeta_q(s)= \sum_{k=1} ^{\infty}q^k \sum_{d\mid k} ^{}d^{s-1}$, $s=1,2,...$, o{\`u} $q$…
We obtain several expansions for $\zeta(s)$ involving a sequence of polynomials in $s$, denoted in this paper by $\alpha_k(s)$. These polynomials can be regarded as a generalization of Stirling numbers of the first kind and our identities…