On the multiple q-Genocchi and Euler numbers

Some New Identities of Genocchi Numbers and Polynomials involving Bernoulli and Euler polynomials

In this paper, we will deal with some new formulae for two product Genocchi polynomials together with both Euler polynomials and Bernoulli polynomials. We get some applications for Genocchi polynomials. Our applications possess a number of interesting properties to study in Theory of Analytic numbers which we express in the present paper.

Recurrence relations for Apostol-Bernoulli , -Euler and -Genocchi polynomials of higher order

In \cite{luo2006,luosri2005}, Luo and Srivastava introduced some generalizations of the Apostol -Bernoulli polynomials and the Apostol-Euler polynomials. The main object of this paper is to extend the result of \cite{prevost2010} to these generalized polynomials. More precisely, using the Padé approximation of the exponential function, we obtain recurrence relations for Apostol-Bernoulli, Euler and also Genocchi polynomials of higher order. As an application we prove lacunary relation for some particular cases.

退化欧拉-塞德尔方法用于退化伯努利、欧拉和戈尼奇多项式

Degenerate Euler- Seidel Method for degenerate Bernoulli, Euler, and Genocchi polynomials

本文提出退化欧拉-塞德尔方法，用于推导退化伯努利、欧拉和戈尼奇多项式的新的组合恒等式。

A unification of the multiple twisted Euler and Genocchi numbers and polynomials associated with p adic q integral on Zp at q=-1

The present paper deals with unification of the multiple twisted Euler and Genocchi numbers and polynomials associated with p-adic q-integral on Zp at q = 1. Some earlier results of Ozden's papers in terms of unification of the multiple twisted Euler and Genocchi numbers and polynomials associated with p-adic q-integral on Zp at q = 1 can be deduced. We apply the method of generating function and p-adic q-integral representation on Zp, which are exploited to derive further classes of Euler polynomials and Genocchi polynomials. To be more precise we summarize our results as follows, we obtain s…

New approach to q-Genocch, Euler numbers and polynomials and their interpolation functions

We give a new construction of q-Genocchi numbers, Euler numbers of higher order, which are different than the q-Genocchi numbers of Cangul-Ozden-Simsek. By using our q-Genoucchi, Euler nimbers of higher order, we can investigate the interesting relationship between w-q-Euler polynomials and w-q-Genocchi polynomials.

A note on the q-Genocchi numbers and polynomials

In ths paper we discuss the new concept of the q-extension of Genocchi numbers and give the some relations between q-Genocchi polynomials and q-Euler numbers.

广义欧拉多项式的一个恒等式

An Identity for Generalized Euler Polynomials

本文提出广义欧拉多项式的一个新恒等式，并借此推广了欧拉数、欧拉多项式、Genocchi多项式与Genocchi数之间的多类经典关系。

A new family of q-analogue of Genocchi numbers and polynomials of higher order

In this work, we consider the generating function of Kim's q-Euler polynomials and introduce new generalization of q-Genocchi polynomials and numbers of higher order. Also, we give surprising identities for studying in Analytic Numbers Theory and especially in Mathematical Physics. Moreover, by applying q-Mellin transformation to generating function of q-Genocchi polynomials of higher order and so we define q-Hurwitz-Zeta type function which interpolates of this polynomials at negative integers.

q Analogue of p adic log gamma functions associated with modified q extension of Genocchi numbers with weight alpha and beta

The fundamental aim of this paper is to describe q-Analogue of p-adic log gamma functions with weight alpha and beta. Moreover, we give relationship between p-adic q-log gamma funtions with weight (α, β) and q-extension of Genocchi numbers with weight alpha and beta and modified q Euler numbers with weight α

Congruences related to Eulerian and Euler numbers

Using Eulerian and Euler numbers, we establish congruences concerning sums involving harmonic numbers, tangent numbers and Genocchi numbers.

p-adic interpolation function related to multiple generalized Genocchi numbers

In the present paper, we deal with multiple generalized Genocchi numbers and polynomials. Also, we introduce analytic interpolating function for the multiple generalized Genocchi numbers attached to \c{hi} at negative integers in complex plane and we define the multiple Genocchi p-adic L-function. Finally, we derive the value of the partial derivative of our multiple p-adic l-function at s=0.

Identities involving the $\left(h,q\right)$-Genocchi polynomials and $\left(h,q\right)$-Zeta-type function

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 $q$-Bernstein polynomials. By considering the $q$-Euler zeta function defined by T. Kim, which can also be obtained by applying the Mellin transformation to the generating function of $\left(h,q\right)$-Genocchi polynomials, we study $\left(h,q\right)$-Zeta-type function. We derive symmetric properties of $\left(h,q\right)$-Zeta function and from these properties we gi…

A note on degenerate multi-poly-Genocchi polynomials

In this paper, we introduce the degenerate multiple polyexponential functions which are multiple versions of the degenerate modified polyexponential functions. Then we consider the degenerate multi-poly-Genocchi polynomials which are defined by using those functions and investigate explicit expressions and some properties for those polynomials.

The median Genocchi numbers, Q-analogues and continued fractions

The goal of this paper is twofold. First, we review the recently developed geometric approach to the combinatorics of the median Genocchi numbers. The Genocchi numbers appear in this context as Euler characteristics of the degenerate flag varieties. Second, we prove that the generating function of the Poincar\' e polynomials of the degenerate flag varieties can be written as a simple continued fraction. As an application we prove that the Poincar\' e polynomials coincide with the $q$-version of the normalized median Genocchi numbers introduced by Han and Zeng.

Arithmetic properties of the Genocchi numbers and their generalization

This note is devoted to establish some new arithmetic properties of the generalized Genocchi numbers $G_{n , a}$ ($n \in \mathbb{N}$, $a \geq 2$). The resulting properties for the usual Genocchi numbers $G_n = G_{n , 2}$ are then derived. We show for example that for any even positive integer $n$, the Genocchi number $G_n$ is a multiple of the odd part of $n$.

New degenerated polynomials arising from non-classical Umbral Calculus

We introduce new generalizations of the Bernoulli, Euler, and Genocchi polynomials and numbers based on the Carlitz-Tsallis degenerate exponential function and concepts of the Umbral Calculus associated with it. Also, we present generalizations of some familiar identities and connection between these kinds of Bernoulli, Euler, and Genocchi polynomials. Moreover, we establish a new analogue of the Euler identity for the degenerate Bernoulli numbers.

On the alternating sums of powers of consecutive q-integers

In this paper we construct $q$-Genocchi numbers and polynomials. By using these numbers and polynomials, we investigate the $q$-analogue of alternating sums of powers of consecutive integers due to Euler.

On variants of the Euler sums and symmetric extensions of the Kaneko-Tsumura conjecture

By using various expansions of the parametric digamma function and the method of residue computations, we study three variants of the linear Euler sums, related Hoffman's double $t$-values and Kaneko-Tsumura's double $T$-values, and establish several symmetric extensions of the Kaneko-Tsumura conjecture. Some special cases are discussed in detail to determine the coefficients of involved mathematical constants in the evaluations. In particular, it can be found that several general convolution identities on the classical Bernoulli numbers and Genocchi numbers are required in this study, and the…

对称函数的新组合恒等式

The new combinatorial identities of symmetric functions

本文利用指数生成函数等初等方法，研究了与对称函数相关的二项式卷积及Bernoulli、Euler和Genocchi数与多项式的卷积和，推导并建立了二元多项式及相关特殊多项式的多个新组合恒等式。

On The Class Of $2D$ $q$-Appell Polynomials

In this research, as the new results of our previously proposed definition for the new class of $2D$ $q$-Appell polynomials, we derive some interesting relations including the recurrence relation and partial $q$-difference equation of the aforementioned family of $q$-polynomials. Next, as some famous examples of this new defined class of $q$-polynomials, we obtain the corresponding relations to the $2D$ $q$-Bernoulli polynomials, $2D$ $q$-Euler polynomials as well as $2D$ $q$-Genocchi polynomials.

欧拉多项式与交替excedances之和

Eulerian polynomials and the alternating sum of excedances

本文通过排列的excedance统计方法，将正切级数系数结构化解释，并建立交替excedances之和与双曲正切函数的关系，揭示了欧拉多项式与切数及Genocchi数的联系。

A note on symmetric properties of the multiple q-Euler zeta functions and higher-order q-Euler polynomials

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 identities concerning the higher-order q-Euler polynomials from the symmetric properties of the multiple q-Euler zeta functions.

q-Euler numbers and polynomials associated with multiple q-zeta functions

The purpose this paper is to present a systemic study of some families of multiple q-Euler numbers and polynomials and we construct multiple q-zeta function which interpolates multiple q-Euler numbers at negative integers.

On a p-adic interpolating function for the multiple generalized Euler numbers and its derivatives

We study analytic function interpolating the multiple generalized Euler numbers attached to $χ$ at negative integers.

Euler-Apéry型多重zeta星值与多重$t$-星值的显式求值

Explicit Evaluation of Euler-Apéry Type Multiple Zeta Star Values and Multiple $t$-Star Values

本文通过迭代积分建立递推关系，进而给出Euler-Apéry型多重zeta星值及其参数变体和多重$t$-星值的显式求值。

分圆欧拉和与多重多对数函数的围道积分及奇偶性结果

Contour Integrations and Parity Results of Cyclotomic Euler Sums and Multiple Polylogarithm Function

该研究通过级数定义扩展三角函数，用围道积分法研究分圆欧拉和及多重多对数函数的奇偶性，得到多类相关显式公式，并推导出经典及交错欧拉和奇偶性的已知公式。

q-Analogue of Euler-Barnes Multiple Zeta Functions (= a note on q-zeta functions)

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 higher order at negative integers.

Integral Evaluation of Odd Euler Sums, Multiple $t$-Value $t\left(3,2,\ldots,2\right)$ and Multiple Zeta Value $ζ(3,2,\ldots,2)$

We construct an analytic approach to evaluate odd Euler sums, multiple zeta value $ζ(3,2,\ldots,2)$ and multiple $t$-value $t\left(3,2,\ldots,2\right)$. Moreover, we also conjecture a closed expression for multiple $t$-value $t\left(2,\ldots,2,1\right)$.

Barnes type multiple q-zeta functions and q-Euler polynomials

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 polynomials of higher order

Multiple $L$-values of level four, poly-Euler numbers, and related zeta functions

We present several formulas for some specific multiple $L$-values of conductor four. This grew out from the study of zeta functions of level four of Arakawa-Kaneko type. Closely related is a new version of multiple poly-Euler numbers and we briefly discuss this too.

Finite Multiple zeta Values and Finite Euler Sums

The alternating multiple harmonic sums are partial sums of the infinite series defining the Euler sums which are the alternating version of the multiple zeta value series. In this paper, we present some systematic structural results of the van Hamme type congruences of these sums, collected as finite Euler sums. Moreover, we relate this to the structure of the Euler sums, which generalizes the corresponding result of the multiple zeta values. We also provide a few conjectures with extensive numerical support.

多重zeta值与Gamma函数的Euler反射公式

Multiple zeta values and Euler's reflection formula for the gamma function

本文利用Hoffman调和代数与二项式恒等式，为源自Gamma函数Euler反射公式的恒等式提供了纯代数证明。

Values at non-positive integers of generalized Euler-Zagier multiple zeta-functions

We give new closed and explicit formulas for "multiple zeta values" at non-positive integers of generalized Euler-Zagier multiple zeta-functions. We first prove these formulas for a small convenient class of these multiple zeta-functions and then use the analyticity of the values on the parameters defining the multiple zeta-functions to deduce the formulas in the general case. Also, for our aim we prove an extension of "Raabe's lemma" due to E. Friedman and A. Pereira.

插值多重ζ值的广义欧拉分解公式

Generalized Euler decomposition formula for interpolated multiple zeta values

本文提出了一种通用的t-shuffle乘积公式，用于推导插值多重ζ值的广义欧拉分解公式，并通过两种方法验证了高度为一的情况。

Alternating multiple zeta values, and explicit formulas of some Euler-Apery-type series

In this paper, we study some Euler-Apéry-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and alternating multiple zeta values. Based on these formulas, we further show that some other series are reducible to ln(2), zeta values, and alternating multiple zeta values by considering the contour integrals related to gamma functions, polygamma functions and trigonometric functions. The evaluations of a large number of special Euler-Apéry-type series are presented as …

Arakawa-Kaneko型与Euler-Zagier型多重zeta函数之间关系的显式公式

Explicit formulas of the relation between multiple zeta functions of Arakawa-Kaneko and Euler-Zagier types

本文给出Arakawa-Kaneko型多重zeta函数表示为Euler-Zagier型多重zeta函数与多重zeta值乘积的Q-线性组合的显式公式，并作为证明关键显式建立multi-polylogarithm函数间的函数方程。

Derivatives of tangent function and tangent numbers

In the paper, by induction, the Faà di Bruno formula, and some techniques in the theory of complex functions, the author finds explicit formulas for higher order derivatives of the tangent and cotangent functions as well as powers of the sine and cosine functions, obtains explicit formulas for two Bell polynomials of the second kind for successive derivatives of sine and cosine functions, presents curious identities for the sine function, discovers explicit formulas and recurrence relations for the tangent numbers, the Bernoulli numbers, the Genocchi numbers, special values of the Euler polyno…

Explicit formulas of Euler sums via multiple zeta values

Flajolet and Salvy pointed out that every Euler sum is a $\mathbb{Q}$-linear combination of multiple zeta values. However, in the literature, there is no formula completely revealing this relation. In this paper, using permutations and compositions, we establish two explicit formulas for the Euler sums, and show that all the Euler sums are indeed expressible in terms of MZVs. Moreover, we apply this method to the alternating Euler sums, and show that all the alternating Euler sums are reducible to alternating MZVs. Some famous results, such as the Euler theorem, the Borwein--Borwein--Girgensoh…

分圆欧拉$T$-和与多重$t$-值的围道积分与奇偶性结果

Contour Integrations and Parity Results of Cyclotomic Euler $T$-Sums and Multiple $t$-Values

本文采用围道积分方法研究非嵌入分圆多重$t$-值（即分圆欧拉$T$-和）的奇偶性，给出了线性与二次情形的显式公式及一般定理，并通过留数计算推导出多个相关公式与推论。

A new Kim's type Bernoulli and Euler Numbers and related identities and zeta and L-functions

In this paper, we consstruct a new extended q-Bernoulli numbers and poly nomials. From these numbers, we derive the multiple zeta functions and give some relations between multiple Bernoulli numbers and multiple zeta functions.

Laurent series expansions of multiple zeta-functions of Euler-Zagier type at integer points

We give explicit expressions (or at least an algorithm of obtaining such expressions) of the coefficients of the Laurent series expansions of the Euler-Zagier multiple zeta-functions at any integer points. The main tools are the Mellin-Barnes integral formula and the harmonic product formulas. The Mellin-Barnes integral formula is used in the induction process on the number of variables, and the harmonic product formula is used to show that the Laurent series expansion outside the domain of convergence can be obtained from that inside the domain of convergence.

On $p$-adic multiple Barnes-Euler zeta functions and the corresponding log gamma functions

Suppose that $ω_1,\ldots,ω_N$ are positive real numbers and $x$ is a complex number with positive real part. The multiple Barnes-Euler zeta function $ζ_{E,N}(s,x;\barω)$ with parameter vector $\barω=(ω_1,\ldots,ω_N)$ is defined as a deformation of the Barnes multiple zeta function as follows $$ ζ_{E,N}(s,x;\barω)=\sum_{t_1=0}^\infty\cdots\sum_{t_N=0}^\infty \frac{(-1)^{t_1+\cdots+t_N}}{(x+ω_1t_1+\cdots+ω_Nt_N)^s}. $$ In this paper, based on the fermionic $p$-adic integral, we define the $p$-adic analogue of multiple Barnes-Euler zeta function $ζ_{E,N}(s,x;\barω)$ which we denote by $ζ_{p,E,N}(…

Explicit Evaluations for Several Variants of Euler Sums

We study several variants of Euler sums by using the methods of contour integration and residue theorem. These variants exhibit nice properties such as closed forms, reduction, etc., like classical Euler sums. In addition, we also define a variant of multiple zeta values of level 2, and give some identities on relations between these variants of Euler sums and the variant of multiple zeta values.

有限与对称欧拉和及有限与对称（交替）多重T-值

Finite and Symmetric Euler Sums and Finite and Symmetric (Alternating) Multiple $T$-Values

本文研究有限多重T-值及其交替版本，利用经典交替MTVs的性质推导出反转、线性洗牌和求和等Q-线性关系，并在计算机辅助下严格计算了小权重情形下这些空间维数的上界。

Computation and theory of Euler sums of generalized hyperharmonic numbers

Recently, Dil and Boyadzhiev \cite{AD2015} proved an explicit formula for the sum of multiple harmonic numbers whose indices are the sequence $\left( {{\left\{ 0 \right\}_r},1} \right)$. In this paper we show that the sums of multiple harmonic numbers whose indices are the sequence $\left( {{\left\{ 0 \right\}_r,1};{\left\{ 1 \right\}_{k-1}}} \right)$ can be expressed in terms of (multiple) zeta values, multiple harmonic numbers and Stirling numbers of the first kind, and give an explicit formula.

Note on multiple q-zeta functions

In this paper, we construct the alternating multiple q-zeta function(= Multiple Euler q-zeta function) and investigate their properties. Finally, we give some interesting functional eauations related to q-Euler polynomials.

Analytic renormalization of multiple zeta functions. Geometry and combinatorics of the generalized Euler reflection formula for MZV

The renormalization of MZV was until now carried out by algebraic means. We show that renormalization in general, of the multiple zeta functions in particular, is more than mere convention. We show that simple calculus methods allow us to compute the renormalized values of multiple zeta functions in any dimension for arguments of the form (1,...,1), where the series do not converge. These values happen to be the coefficients of the asymptotic expansion of the inverse Gamma function. We focus on the geometric interpretation of these values, and on the combinatorics their closed form encodes, wh…

Multiple zeta values and Euler sums

In this paper, we establish some expressions of series involving harmonic numbers and Stirling numbers of the first kind in terms of multiple zeta values, and present some new relationships between multiple zeta values and multiple zeta star values. The relationships obtained allow us to find some nice closed form representations of nonlinear Euler sums through Riemann zeta values and linear sums. Furthermore, we show that the combined sums \[H\left( {a,b;m,p} \right) := \sum\limits_{a + b = m - 1} {ζ\left( {{\left\{ p \right\}_a},p + 1,{\left\{ p \right\}_b}} \right)}\quad (m\in \N,p>1) \] an…

On Generalized Multi Poly-Euler and Multi Poly-Bernoulli Polynomials

In this paper, we establish more identities of generalized multi poly-Euler polynomials with three parameters and obtain a kind of symmetrized generalization of the polynomials. Moreover, generalized multi poly-Bernoulli polynomials are defined using multiple polylogarithm and derive some properties parallel to those of poly-Bernoulli polynomials. These are generalized further using the concept of Hurwitz-Lerch multiple zeta values.

Identities of symmetry for (h;q)-extension of higher-order Euler polynomials

In this paper, we study some symmetric properties of the multiple q-Euler zeta function. From these properties, we derive several identities of symmetry for the (h;q)-extension of higher-order Euler polynomials.

Some identities of symmetry for the generalized q-Euler polynomials

By the symmetric properties of Drichlet's type multiple q-l-function, we establish various identities concerning the generalized higher-order q-Euler polynomials. Furthermore, we give some interesting relationship between the power sums and the generalized higher-order $q$-Euler polynomials.

Analytic properties of generalized Mordell-Tornheim type of multiple zeta-functions and L-functions

Analytic properties of three types of multiple zeta functions, that is, the Euler-Zagier type, the Mordell-Tornheim type and the Apostol-Vu type have been studied by a lot of authors. In particular, in the study of multiple zeta functions of the Apostol-Vu type, a generalized multiple zeta function, including both the Euler-Zagier type and the Apostol-Vu type, was introduced.In this paper, similarly we consider generalized multiple zeta-functions and $ L $-functions, which include both the Euler-Zagier type and the Mordell-Tornheim type as special cases.We prove the meromorphic continuation to…

Identities of symmetry for higher-order generalized q-Euler polyonmials

In this paper, we investigate the properties of symmetry in two variables related to multiple Euler q-l-function which interpolates higher-order q-Euler polynomials at negative integers. From our investigation, we can derive many interesting identities of symmetry in two variables related to generalized higher-order q-Euler polynomials and alternating generalized q-power sums.

$q$-Volkenborn Integration and Its Applications

The main purpose of this paper is to present a systemic study of some families of multiple $q$-Euler numbers and polynomials. In particular, by using the $q$-Volkenborn integration on $\Bbb Z_p$, we construct $p$-adic $q$-Euler numbers and polynomials of higher order. We also define new generating functions of multiple $q$-Euler numbers and polynomials. Furthermore, we construct Euler $q$-Zeta function. This paper is the version of the preprint of my paper to publish in the special issues as the preprint part of this proceedings.