A q-analog of Euler's decomposition formula for the double zeta function

This work is an example driven overview article of recent works on the connection of multiple zeta values, modular forms and q-analogues of multiple zeta values given by multiple Eisenstein series.

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 always recover the corresponding classical identities by taking $q\to 1$. The main result of the paper is the duality relations between multiple zeta star values and Euler sums and their $q$-analogs, which are generalizations of the Two-one f...

We study the twisted q-zeta functions and twisted q-Bernoulli polynomials

In this paper, we investigate an asymptotic behavior of the double zeta function of Euler-Zagier type for indices with large negative real parts.

This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers $H_{n}^{\left( p,q\right) }$ \[ \zeta_{H^{\left( p,q\right) }}\left( r\right) =\sum\limits_{n=1}^{\infty }\dfrac{H_{n}^{\left( p,q\right) }}{n^{r}}% \] in terms of the famous Euler sums of generalized harmonic numbers. Moreover, several infinite series, whose terms consist of certain harmonic numbers and reciprocal binomial coefficients, are evaluated in terms of Riemann zeta values.

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 representation for the general multiple polylogarithm, and develop a q-analogue of the shuffle product.

In this short and elementary note we derive a q-generalization of Euler's decomposition formula for the qMZVs recently introduced by Y. Ohno, J. Okuda, and W. Zudilin. This answers a question posed by these authors in [10].

In this paper we define the generalized q-analogues of Euler sums and present a new family of identities for q-analogues of Euler sums by using the method of Jackson q-integral rep- resentations of series. We then apply it to obtain a family of identities relating quadratic Euler sums to linear sums and q-polylogarithms. Furthermore, we also use certain stuffle products to evaluate several q-series with q-harmonic numbers. Some interesting new results and illustrative example...

Thw purpose of this paper is to present a systemic study of some families of the generalized q-Euler numbers and polynomials of higher order.

A $q$-analogue of the multiple gamma functions is introduced, and is shown to satisfy the generalized Bohr-Morellup theorem. Furthermore we give some expressions of these function.