q-analogue of Euler-Barnes' numbers and polynomials

On the ratios of Barnes' multiple gamma functions to the $p$-adic analogues

Let $F$ be a totally real field. For each ideal class $c$ of $F$ and each real embedding $ι$ of $F$, Hiroyuki Yoshida defined an invariant $X(c,ι)$ as a finite sum of log of Barnes' multiple gamma functions with some correction terms. Then the derivative value of the partial zeta function $ζ(s,c)$ has a canonical decomposition $ζ'(0,c)=\sum_ιX(c,ι)$, where $ι$ runs over all real embeddings of $F$. Yoshida studied the relation between $\exp(X(c,ι))$'s, Stark units, and Shimura's period symbol. Yoshida and the author also defined and studied the $p$-adic analogue $X_p(c,ι)$: In particular, we di…

函数方程、上界以及Barnes双ζ函数的Lindelöf假设类比

Functional equation, upper bounds and analogue of Lindelöf hypothesis for the Barnes double zeta function

本文研究Barnes双ζ函数的函数方程，并利用Phragmén-Lindelöf凸性原理，推导出该函数在t趋于无穷时的上界。

Barnes多重zeta函数及相关问题的Lindelöf假设类比

Analogues of the Lindelöf Hypothesis for the Barnes multiple zeta function and related problems

研究Barnes多重zeta函数的Lindelöf假设类比，通过积分均值建立等价条件，并揭示依赖于有理秩的独特现象。

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.

Relations for Bernoulli--Barnes Numbers and Barnes Zeta Functions

The \emph{Barnes $ζ$-function} is \[ ζ_n (z, x; \a) := \sum_{\m \in \Z_{\ge 0}^n} \frac{1}{\left(x + m_1 a_1 + \dots + m_n a_n \right)^z} \] defined for $\Re(x) > 0$ and $\Re(z) > n$ and continued meromorphically to $\C$. Specialized at negative integers $-k$, the Barnes $ζ$-function gives \[ ζ_n (-k, x; \a) = \frac{(-1)^n k!}{(k+n)!} \, B_{k+n} (x; \a) \] where $B_k(x; \a)$ is a \emph{Bernoulli--Barnes polynomial}, which can be also defined through a generating function that has a slightly more general form than that for Bernoulli polynomials. Specializing $B_k(0; \a)$ gives the \emph{Bernoul…

A note on the sums of powers of consecutive q-integers

In this paper we construct the q-analogue of Barnes' Bernoulli numbers and plynomials of degree 2, which is an answer to a part of Schlosser's question. Finally, we treat the q-analogue of the sums of powers of consecutive integrs.

A $(p,q)$-Analogue of Poly-Euler Polynomials and Some Related Polynomials

In the present article, we introduce a $(p,q)$-analogue of the poly-Euler polynomials and numbers by using the $(p,q)$-polylogarithm function. These new sequences are generalizations of the poly-Euler numbers and polynomials. We give several combinatorial identities and properties of these new polynomials. Moreover, we show some relations with the $(p,q)$-poly-Bernoulli polynomials and $(p,q)$-poly-Cauchy polynomials. The $(p,q)$-analogues generalize the well-known concept of the $q$-analogue.

On q-analogues of quadratic Euler sums

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 examples are considered. Finally, we can obtain some explicit relations for the classical Euler sums when q approaches to 1.

关于Barnes双ζ函数的Laurent级数展开

On the Laurent series expansions of the Barnes double zeta function

本文研究Barnes双ζ函数在s=1和s=2处的Laurent级数展开，推导出系数的极限表达式，得到类似于欧拉-斯蒂尔杰斯常数的类比，并分析其渐进行为，揭示其与经典理论的区别。

A $q$-analogue for Euler's evaluations of the Riemann zeta function

We provide a $q$-analogue of Euler's formula for $ζ(2k)$ 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 $ζ(2)=π^2/6$ and $ζ(4)=π^4/90$.

A $q$-analogue for Euler's $ζ(6)=π^6/945$

We give a $q$-analogue of $ζ(6)=π^6/945$. Our main results are stated in Theorems 2.1 and 2.2 below.

关于迪比茨公式和欧拉常数的有限类比及其格雷戈里多项式

On finite analogues of Dobiński's formula and of Euler's constant via Gregory polynomials

本文研究了与欧拉常数e相关的迪比茨公式的有限类比及其贝塞尔型推广，并利用格雷戈里多项式扩展了kaneko-matsusaka-seki关于欧拉常数有限类比的研究，与wilson型类比γ_a^w进行比较。

Divisor Problem and an Analogue of Euler's Summation Formula

By simple elementary method,we obtain with ease,a highly simple expression for the remainder term of the divisor problem and use it to obtain an Euler-Maclaurin analogue of summation involving divisor function.We also obtain a relation connecting the remainder term of the divisor problem and the remainders of approximate functional equations for Riemann zeta function and its square,for positive values of arguments.

Analogues of Euler and Poisson summation formulae

Euler--Maclaurin and Poisson analogues of the summations $\sum_{a < n \leq b} χ(n) f(n)$, $\sum_{a < n \leq b} d(n) f(n)$, $\sum_{a < n \leq b} d(n) χ(n) f(n)$ have been obtained in a unified manner, where $(χ(n))$ is a periodic complex sequence; $d(n)$ is the divisor function and $f(x)$ is a sufficiently smooth function on $[a,b]$. We also state a generalised Abel's summation formula, generalised Euler's summation formula and Euler's summation formula in several variables.

Note on the sums of powers of consecutive $q$-integers

In this paper we construct the $q$-analogue of Barnes's Bernoulli numbers and polynomials of degree 2, for positive even integers, which is an answer to a part of Schlosser's question. For positive odd integers, Schlosser's question is still open. Finally, we will treat the $q$-analogue of the sums of powers of consecutive integers.

关于欧拉常数的有限模拟

On finite analogues of Euler's constant

本文在有限多重zeta值的框架下引入欧拉常数的有限模拟，从正则化ζ(1)及级数表达式定义候选值，并证明其差异属于由1和对数有限模拟值张成的有理向量空间。

Note on q-extensions of Euler numbers and polynomials of higher order

In [14] Ozden-Simsek-Cangul constructed generating functions of higher-order twisted $(h,q)$-extension of Euler polynomials and numbers, by using $p$-adic q-deformed fermionic integral on $\Bbb Z_p$. By applying their generating functions, they derived the complete sums of products of the twisted $(h,q)$-extension of Euler polynomials and numbers, see[13, 14]. In this paper we cosider the new $q$-extension of Euler numbers and polynomials to be different which is treated by Ozden-Simsek-Cangul. From our $q$-Euler numbers and polynomials we derive some interesting identities and we construct $q…

q-Euler and Genocchi numbers

Carlitz has introduced an interesting $q$-analogue of Frobenius-Euler numbers in [4]. He has indicated a corresponding Stadudt-Clausen theorem and also some interesting congruence properties of the $q$-Euler numbers. In this paper we give another construction of $q$-Euler numbers, which are different than his $q$-Euler numbers. By using our $q$-Euler numbers, we define the $q$-analogue of Genocchi numbers and investigate the relations between $q$-Euler numbers and $q$-analogs of Genocchi numbers.

欧拉双重zeta值的$q$-模拟之间的代数恒等式

Algebraic identities among $q$- analogue of Euler double zeta values

本文探究另一类q-双重zeta函数变体及其星型变体的恒等式，借助q>1的Nielsen反射公式q-模拟，研究相关恒等式，分析参数趋于0的行为并与经典情形对比，还讨论Mordell-Tornheim r重zeta函数q-模拟及其与q-双重zeta函数的关系。

欧拉常数的多对数类似物

Polylogarithmic Analogues of Euler's Constant

引入一族常数 C_m，作为欧拉常数的多对数类似物，研究其性质并建立与伽马函数相关的迭代对数积分表示，进一步引入多对数zeta势和多对数伽马函数，并应用于将C_m与涉及黎曼zeta函数的狄利克雷级数的特殊值联系起来。

非阿贝尔扩展中的素数分解及其与欧拉准则的类比

The decomposition of primes in nonabelian extensions of Heisenberg type and an analogue of Euler's criterion

本文研究非阿贝尔扩展中素数的分解特性，通过显式多项式条件确定一次素数的完全分解，与欧拉准则有类比关系。

Identities of symmetry for q-Euler polynomials

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 about identities of symmetry in two variables. These abundance of symmetries shed new light even on the existing identities so as to yield some further interesting ones. The derivations of identities are based on the $p$-adic integral expression of the generating function for the $q$-Euler polynomials and the quotient of integrals that can be expressed as th…

Note on q-Dedekind type sums related to q-Euler polynomials

it is the purpose of this paper to construct a p-adic continuous function for an odd prime to contain a p-adic q-analogue of higher order Dedekind type sums related to q-Euler polynomials and numbers.

Some identities of Frobenius-Euler polynomials arising from Frobenius-Euler basis

In this paper, we give some new and interesting identities which are derived from the basis of Frobenius-Euler. Recently, Simsek et als(see [13]) have given some identities of q-analogue of Frobenius-Euler polynomials related to q-Bernstein polynomials. From the methods of our paper, we can also derive the results and identities of Simsek et als (cf.[13]).

Two $q$-analogues of Euler's formula $ζ(2)=π^2/6$

It is well known that $ζ(2)=π^2/6$ as discovered by Euler. In this paper we present the following two $q$-analogues of this celebrated formula: $$\sum_{k=0}^\infty\frac{q^k(1+q^{2k+1})}{(1-q^{2k+1})^2}=\prod_{n=1}^\infty\frac{(1-q^{2n})^4}{(1-q^{2n-1})^4}$$ and $$\sum_{k=0}^\infty\frac{q^{2k-\lfloor(-1)^kk/2\rfloor}}{(1-q^{2k+1})^2} =\prod_{n=1}^\infty\frac{(1-q^{2n})^2(1-q^{4n})^2}{(1-q^{2n-1})^2(1-q^{4n-2})^2},$$ where $q$ is any complex number with $|q|<1$. We also give a $q$-analogue of the identity $ζ(4)=π^4/90$, and pose a problem on $q$-analogues of Euler's formula for $ζ(2m)\ (m=3,4,\l…

Identities of symmetry for higher-order q-Euler polynomials

In this paper, we derive basic identities of symmetry in two variables related to higher-order q-Euler polynomials and q-analogue of higher order alternating power sums. The derivation of identities are based on the multibvariate p-adic fermionic integral espression of the generating function for the higher-order q-Euler polynomials.

三次阿贝尔域的Euler-Kronecker常数的极值

Extreme values of Euler-Kronecker constants of abelian cubic fields

本文在广义黎曼猜想下，研究三次阿贝尔域和二次域的Euler-Kronecker常数的极值，给出了相应结果。

有限域上多项式的Euler totient函数研究

On Euler's totient function of polynomials over finite fields

该研究聚焦有限域上多项式的Euler totient函数的算术性质，分析多个经典数论猜想的多项式类比形式，拓展了有限域多项式数论的研究方向。

On the q-Analogue of Gamma Functions and Related Inequalities

In this, paper we obtain a q-analogue of a double inequality involving the Euler gamma function which was first proved geometrically by Alsina and Tomas and then analytically by Sandor

The fermionic p-adic integrals on Zp associated with extended q-Euler numbers and polynomials

The purpose of this paper is to present a systemic study of some families of q-Euler numbers and polynomials of Norlund's type by using multivariate fermionic p-adic integral on Zp. Moreover, the study of these higher-order q-Euler numbers and polynomials implies some interesting q-analogue of stirling numbers identities.

q-Euler Numbers and Polynomials Associated with Basic Zeta Functions

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 integers. Finally we woll treat some identities of the q-extension of the euler numbers by using fermionic p-adic q-integration on Z_p.

p-adic interpolating function associated with modified Dirichlet's type of twisted q-euler numbers and polynomials with weight alpha

The q-calculus theory is a novel theory that is based on finite difference re-scaling. The rapid development of q-calculus has led to the discovery of new generalizations of q-Euler polynomials involving q-integers. The present paper deals with the modified Dirichlet's type of twisted q-Euler polynomials with weight alpha. We apply the method of generating function and p-adic q-integral representation on Zp, which are exploited to derive further classes of q-Euler numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between mod…

Hypergeometric Euler numbers

In this paper, we introduce the hypergeometric Euler number as an analogue of the hypergeometric Bernoulli number and the hypergeometric Cauchy number. We study several expressions and sums of products of hypergeometric Euler numbers. We also introduce complementary hypergeometric Euler numbers and give some characteristic properties. There are strong reasons why these hypergeometric numbers are important. The hypergeometric numbers have one of the advantages that yield the natural extensions of determinant expressions of the numbers, though many kinds of generalizations of the Euler numbers h…

Extended q-Dedekind-type Daehee-Changhee sums associated with Extended q-Euler polynomials

In the present paper, our goal is to introduce a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order dedekind-type sums with weight alpha related to Extended q-Euler polynomials by using p-adic q-integral in the p-adic integer ring.

$q$-Analogues of some supercongruences related to Euler numbers

Let $E_n$ be the $n$-th Euler number and $(a)_n=a(a+1)\cdots (a+n-1)$ the rising factorial. Let $p>3$ be a prime. In 2012, Sun proved the that $$ \sum^{(p-1)/2}_{k=0}(-1)^k(4k+1)\frac{(\frac{1}{2})_k^3}{k!^3} \equiv p(-1)^{(p-1)/2}+p^3E_{p-3} \pmod{p^4}, $$ which is a refinement of a famous supercongruence of Van Hamme. In 2016, Chen, Xie, and He established the following result: $$ \sum_{k=0}^{p-1}(-1)^k (3k+1)\frac{(\frac{1}{2})_k^3}{k!^3} 2^{3k} \equiv p(-1)^{(p-1)/2}+p^3E_{p-3} \pmod{p^4}, $$ which was originally conjectured by Sun. In this paper we give $q$-analogues of the above two supe…

关于多元多项式系统的欧拉φ函数的推广

Generalizations of Euler's $φ$-function with respect to systems of polynomials of several variables

该研究引入多元多项式系统相关的欧拉φ函数新推广，用直接方法证恒等式、研特殊情况，还探讨酉类似物，为相关领域研究提供新内容与思路。

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

A variation of Euler's approach to values of the Riemann zeta function

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 limit of this q-analogue exists and equals the value of the Riemann zeta.

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.

A note on q-Volkenborn integration

In this paper, we construct the new $q$-analogue of the ordinary Euler numbers and polynomials by using the $q$-Volkenborn integrals.

An analogue of Kummer's relation between the ideal class number and the unit index of cyclotomic fields

In this paper, we obtain a formula for the special value of Euler-Dirichlet $L$-function $L_E(s,χ)$ at $s=1$. This leads to another class number formula of $\mathbb{Q}(μ_{m})^{+}$, the maximal real subfield of $m$th cyclotomic field. From this formula, we construct a new type of cyclotomic units in $\mathbb{Q}(μ_{p^{n}})$, which implies a similar Kummer's relation between the ideal class number of $\mathbb{Q}(μ_{p^{n}})^{+}$ and the unit index.

Euler Type Generalization of Wilson's Theorem

In this short note, we introduce an Euler analogue of Wilson's theorem; $a_1a_2... a_{ϕ(n)}\equiv (-1)^{ϕ(n)+1}~({\rm mod}~n)$ say, where ${\rm gcd}(a_i,n)=1$.

一些分拆结果的固定周长类比

Fixed perimeter analogues of some partition results

本文受 Euler 奇偶分拆恒等式及 Straub 固定周长结果启发，研究若干经典分拆结论在固定最大 hook（周长）条件下的类比形式。

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.

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.

A Polynomial Variation on Meinardus' Theorem

We develop a polynomial analogue of Meinardus' Thoerem for bivariate Euler products and apply it to the study of complex multiplicatively weighted partitions.

Van Hamme 若干同余式的一个精细 q-模拟

A refined $q$-analogue of some congruences of Van Hamme

本文受 Guo 近期工作启发，为 Van Hamme 的 (B.2)、(E.2)、(F.2) 超同余式建立了统一的 q-模拟，并由此导出与 Euler 多项式相关的含参数超同余式。