Related papers: Relations for Bernoulli--Barnes Numbers and Barnes…

The functional equations of the Riemann zeta function, the Hurwitz zeta function, and the Lerch zeta function have been well known for a long time, and there is great importance in studying these zeta functions. For example, fundamental…

The main purpose of this paper is to show some relations between the Riemann zeta function and the generalized Bernoulli polynomials of level $m$. Our approach is based on the use of Fourier expansions for the periodic generalized Bernoulli…

The fundamental objective of this paper is to obtain some interesting properties for $\left(h,q\right)$-Genocchi numbers and polynomials by using the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ and mentioned in the paper…

We study the interplay between recurrences for zeta related functions at integer values, `Minor Corner Lattice' Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and…

In this work we introduce a new polynomial representation of the Bernoulli numbers in terms of polynomial sums allowing on the one hand a more detailed understanding of their mathematical structure and on the other hand provides a…

The main aim of this paper is to investigate and introduce relations between the numbers of k-ary Lyndon words and unified zeta-type functions which was defined by Ozden et al [15, p. 2785]. Finally, we give some identities on generating…

Suppose that $\omega_1,\ldots,\omega_N$ are positive real numbers and $x$ is a complex number with positive real part. The multiple Barnes-Euler zeta function $\zeta_{E,N}(s,x;\bar\omega)$ with parameter vector…

It is known that the special values of multiple zeta functions at non-positive arguments are indeterminate in most cases due to the occurrences of infinitely many singularities. In order to give a suitable rigorous meaning of the special…

We consider the variation of two fundamental types of zeta functions that arise in the study of both physical and analytical problems in geometric settings involving conical singularities. These are the Barnes zeta functions and the Bessel…

In his second notebook, Ramanujan discovered the following identity for the special values of $\zeta(s)$ at the odd positive integers \begin{equation*}\begin{aligned}\alpha^{-m}\,\left\{\dfrac{1}{2}\,\zeta(2m + 1) + \sum_{n =…

In the former part of this paper, we give functional equations for Barnes multiple zeta-functions and consider some relevant results. In particular, we show that Ramanujan's classical formula for the Riemann zeta values can be derived from…

This paper is devoted to Ser's and Hasse's series representations for the zeta-functions, as well as to several closely related results. The notes concerning Ser's and Hasse's representations are given as theorems, while the related…

Building on the mapping relations between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, and by defining the Zeta and related functions including the Hurwitz Zeta function…

In 1999, Arakawa and Kaneko introduced a zeta function whose special values at negative integers yield the poly-Bernoulli numbers and investigated its relation to multiple zeta values. Since the poly-Bernoulli numbers appear in this…

The aim of this paper is to define some new number-theoretic functions including necklaces polynomials and the numbers of special words such as Lyndon words. By using Dirichlet convolution formula with well-known number-theoretic functions,…

We construct the new q-extension of Bernoulli numbers and polynomials in this paper. Finally we consider the q-zeta functions which interpolate the new q-extension of Bernoulli numbers and polynomials.

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…

The Barnes multiple zeta function is useful to study in the number theory and Knot thoey and Mathematical Physics. In this paper we consider q-extension of Barnes type multiple zeta function and we also construct the q-extension of Euler…

In this paper, we introduce $q$-analogues of the Barnes multiple zeta functions. We show that these functions can be extended meromorphically to the whole plane, and moreover, tend to the Barnes multiple zeta functions when $q\uparrow 1$…

We introduce Bernoulli-Dunkl and Apostol-Euler-Dunkl polynomials as generalizations of Bernoulli and Apostol-Euler polynomials, where the role of the derivative is now played by the Dunkl operator on the real line. We use them to sum a…