Nabiullah Khan, Saddam Husain
In this paper, we present a new type of generating function of generalized Apostol-Bernoulli poly Daehee polynomial (GABPDP). By using the generating function of GABPDP, we discuss some special cases and useful identities of generalized Apostol-Bernoulli poly Daehee polynomials. We also drive implicit summation formulae of it.
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The main purpose of this paper is to present various identities and computation formulas for certain classes of Apostol-type numbers and polynomials. The results of this paper contain not only the $λ$-Apostol-Daehee numbers and polynomials, but also Simsek numbers and polynomials, the Stirling numbers of the first kind, the Daehee numbers, and the Chu-Vandermonde identity. Furthermore, we derive an infinite series representation for the $λ$-Apostol-Daehee polynomials. By using functional equations containing the generating functions for the Cauchy numbers and the Riemann integrals of the gener…
By applying the p-adic q-Volkenborn Integrals including the bosonic and the fermionic p-adic integrals on p-adic integers, we define generating functions, attached to the Dirichlet character, for the generalized Apostol-Bernoulli numbers and polynomials, the generalized Apostol-Euler numbers and polynomials, generalized Apostol-Daehee numbers and polynomials, and also generalized Apostol-Changhee numbers and polynomials. We investigate some properties of these numbers and polynomials with their generating functions. By using these generating functions and their functional equation, we give som…
Higher-order Daehee numbers and polynomials
最近，作者引入了Daehee数和多项式。本文考虑了Daehee数和k阶多项式，给出了k阶Daehee多项式与特殊多项式之间的一些关系
In this paper, we consider the degenerate Daehee numbers and polynomials of the second kind which are different from the previously introduced Daehee numbers and polynomials. We investigate some properties of these numbers and polynomials. In addition, we give some new identities and relations between the Daehee polynomials of the second kind and Carlitz's degenerate Bernoulli polynomials.
Daehee Numbers and polynomials
我们考虑了Daehee数和多项式的Witt型公式，并研究了这些数和多项式的一些性质。特别是，Daehee数与高阶Bernoulli数和第二类Bernoulli数密切相关。
In this paper, we study a degenerate version of the Daehee polynomials and numbers, namely the degenerate Daehee polynomials and numbers, which were recently introduced by Jang et. al. We derive their explicit expressions and some identities involving them. Further, we introduce the multiple degenerate Daehee numbers and higher-order degenerate Daehee polynomials and numbers which can be represented in terms of integrals on the unitcube. Again, we deduce their explicit expressions and some identities related to them.
Catalan-Daehee numbers and polynomials, generating functions of which can be expressed as p-adic Volkenborn integrals on Zp, were studied previously. The aim of this paper is to introduce q-analogues of the catalan-Daehee numbers and polynomials with the help of p-adic q-integrals von Zp. We derive, among other thing, some explicit expressions for the q-analogues of the Catalan-Daehee numbers and polynomials.
Barnes-type Daehee polynomials
本文考虑Barnes型第一类和第二类Daehee多项式。从这类多项式的Sheffer序列的性质出发，我们得到了一些新的有趣的恒等式。
Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials
这是Tom Apostol介绍的一类函数的简短调查。调查的重点是他们的关系欧拉多项式，导数多项式，也对一些整表示。
In recent years, many authors have studied Changhee and Dae- hee polynomials in connection with many special numbers and polynomials. In this paper, we investigate type 2 Changhee and Daehee numbers and polynomials and give some identities for these numbers and polynomials in relation to type 2 Euler and Bernoulli numbers and polynomials. In addition, we express the central factorial numbers of the second kind in terms of type 2 Bernoulli, type 2 Changhee and type 2 Daehee numbers of negative integral orders.
In this paper we consider the Daehee numbers and polynomials of the first and second kind, and give several explicit representations for them. In particular, we express the Daehee polynomials as the derivative of a generalized binomial coefficient. This is done by performing the Stirling transform of the power sum polynomial $S_k(x)$ associated with the sum of $k$th powers of the first $n$ positive integers $S_k(n) = 1^k + 2^k + \cdots + n^k$. Furthermore, we show the relationship between the Daehee polynomials and the hyperharmonic polynomials. This allows us to also express the hyperharmonic…
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 bunch of series involving the zeros of Bessel functions.
Extended q-Dedekind-type Daehee-Changhee sums associated with Extended q-Euler polynomials
本文利用p-adic整数环上的p-adic q-积分，在与推广的q-欧拉多项式有关的带权α的高阶dedekind型和的p-adic q-模拟中引入了一个奇素数的p-adic连续函数.
A note on the q-Dedekind-type Daehee-Changhee sums with weight alpha arising from modified q-Genocchi polynomials with weight alpha
本文利用Zp上的p-adic不变q-积分，讨论了高阶带权Dedekind型和的p-adic q-模拟内的p-adic奇素数连续函数与带权α的修正q-Genocchi多项式的关系.
In this note, we shall obtain two closed forms for the Apostol-Bernoulli polynomials.
In this paper, by introducing the degenerate Fubini-type polynomials, we give several relations with the help of the Faà di Bruno formula and some properties of Bell polynomials, and generating function methods. Also, we derive some new explicit formulas and recurrence relations for Fubini-type polynomials and numbers. Associating the degenerate Fubini-type polynomials newly defined here with degenerate Apostol-Bernoulli polynomials and degenerate Apostol-Euler polynomials of order α enables us to present additional relations for some degenerate special polynomials and numbers.
Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials
详细分析了伯努利多项式$\mathcal{B}_{n}(x;λ)$的渐近性态.起点是他们的傅里叶级数的$[0,1]$，它示出，仍然有效的渐近展开在紧凑的子集的复平面。这是用来确定明确的估计常数的近似，并分析振荡现象，在某些情况下出现。这些结果被转移到Apostol-欧拉多项式$\mathcal{E}_{n}(x;λ)$通过一个简单的关系连接到Apostol-伯努利多项式。
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.
Some Results for the Apostol-Genocchi Polynomials of Higher Order
本文研究了高阶Apostol-Genocchi多项式的乘法公式，并导出了一些显式递推公式。可以推出Carlitz和Howard关于Genocchi数的一些较早的结果。我们引入二元Apostol-Genocchi多项式，然后我们考虑二元Genocchi多项式的乘法定理。引入了带a; b; c参数的广义Apostol-Genocchi多项式，得到了关于带a; b; c参数的广义Apostol-Genocchi多项式的几个恒等式
Apostol-Euler polynomials arising from umbral calculus
本文利用投影演算中定义的正交型，导出了几个著名多项式作为Apostol-欧拉多项式的线性组合的显式公式。
Some new identities on the Apostol-Bernoulli polynomials of higher order derived from Bernoulli basis
本文利用伯努利多项式基导出了高阶Apostol-伯努利多项式的新的有趣关系和恒等式.最后，利用我们的方法，我们还推出了公式的卷积的伯努利和欧拉多项式，表示为Apostol-伯努利多项式的高阶。
Many authors studied mix type polynomials and their properties. For their numerous uses in Statistics, Combinatorial analysis and Number theory. In the present article, we define a generating function of mix type Bell based Apostol type polynomials and numbers of order a. Furthermore, we derive some elementary properties and identities of Bell based Apostol type polynomials of order a such as correlation formulas, Implicit summation formulas, derivative formulas and their special cases. Also, we derive some relation with Stirling numbers.
We prove the simultaneous multiplication formulas for Apostol-Bernoulli polynomials and generalized Frobenius-Euler polynomials. These formulas contain Dedekind-Rademacher sums, Apostol-Dedekind sums and Fourier-Dedekind sums.
In this paper we evaluate sums and integrals of products of Fubini polynomials and have new explicit formulas for Fubini polynomials and numbers. As a consequence of these results new explicit formulas for p-Bernoulli numbers and Apostol-Bernoulli functions are given. Besides, integrals of products of Apostol-Bernoulli functions are derived.
The object of this paper is to introduce and study properties of unified Apostol-Bernoulli and Apostol-Euler polynomials noted by $\left\{\mathfrak{V_{n}}(x;λ;μ)\right\}_{n \geq 0}$. We study some arithmetic properties of $\left\{\mathfrak{V_{n}}(x;λ;μ)\right\}_{n \geq 0}$ as their connection to Apostol-Euler polynomials and Apostol-Bernoulli polynomials. Also, we give derivation and integration representations of $\left\{\mathfrak{V_{n}}(x;λ;μ)\right\}_{n \geq 0}$. Finally, we use the umbral calculus approach to deduce symmetric identities.
The elliptic Apostol-Dedekind sums generate odd Dedekind symbols with Laurent polynomial reciprocity laws
Dedekind符号是经典Dedekind和（符号）的推广。具有Laurent多项式互反律的Dedekind符号空间与模形式空间之间存在自然同构。我们将定义一个新的椭圆模拟的Apostol-Dedekind和。然后，我们将表明，新定义的和生成所有奇Dedekind符号与劳伦多项式互易律。我们的构造是基于Machide关于椭圆型Dedekind-Rademacher和的结果。作为我们结果的应用，我们发现了艾森斯坦系列等式，这些等式推广了Ramanujan、van der Pol、Rankin和Skoruppa的某些公式。
Exponential generating functions for Apostol-Dedekind and related sums
本文构造 Apostol-Dedekind 和的指数生成函数，定义高阶和及相关有限和族，并利用这些函数推导出许多新公式。
In this paper, we give various identities for the weighted average of the product of generalized Anderson-Apostol sums with weights concerning completely multiplicative function, completely additive function, logarithms, the Gamma function, Bernoulli polynomials and binomial coefficients respectively. Our results generalize many known results.
On the associated sequences of special polynomials
本文研究了Daehee多项式和Changhee多项式的相伴序列的一些性质。最后，我们给出了涉及某些特殊多项式的相伴序列的一些有趣的恒等式。
On the special values of Berndt's generalized $L$-function
本文研究Berndt广义L函数，引入广义Apostol-Bernoulli多项式与函数并建立其性质，证明该L函数整数点值可通过上述函数显式计算。
Ramaswami Type translation formulae for the polylogarithm functions
本文研究多项式对数函数的Ramaswami和Apostol类型翻译公式，并探讨其在伯努利数递推关系和切线数同余性质中的应用。
Recently, the numbers $Y_{n}(λ)$ and the polynomials $Y_{n}(x,λ)$ have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating functions. By using these generating functions with their functional equations and derivative equations, we derive various identities and relations including two recurrence relations, Vandermonde type convolution formula, combinatorial sums, the Bernstein basis functions, and also some well known families of special numbers and their interpolation functions such as the Apos…
The classical Dedekind sums appear in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. The Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations, and are shown to satisfy some reciprocity relations. In contrast, Dedekind type DC (Daehee and Changhee) sums and their generalizations are defined in terms of Euler functions and their generalizations. The purpose of this paper is to introduce the poly-Dedekind type DC sums, which are obtained from the Dedekind type DC sums by…
We consider the class $\mathcal{E}_t(Y)$ of Appell polynomials whose generating function is given by means of a real power $t$ of the moment generating function of a certain random variable $Y$. For such polynomials, we obtain explicit expressions depending on the moments of $Y$. It turns out that various kinds of generalizations of Bernoulli and Apostol-Euler polynomials belong to $\mathcal{E}_t(Y)$ and can be written and investigated in a unified way. In particular, explicit expression for such polynomials can be given in terms of suitable probabilistic generalizations of the Stirling number…
Dedekind sums occur in the transformation behaviour of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider poly-Dedekind sums which are obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and pr…
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 functions for the numbers of k-ary Lyndon words and some special numbers and polynomials such as the Apostol-Bernoulli numbers and polynomials, Frobenius-Euler numbers, Euler numbers and Bernoulli numbers.
The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study contains both new results and literature surveys about some of the related results that have existed so far. With the aid of hypergeometric function, generating functions for a new family of the combinatorial numbers, related to finite sums of powers of binomial coefficients, are constructed. By using these generating functions, a number of new identities have…
The computation of $ζ(2k)$, $β(2k+1)$ and beyond by using telescoping series
本文利用叠缩级数和 Bernoulli、Euler 多项式的定义性质，给出 ζ(2k)、β(2k+1) 的简单证明，并推广到 ζ(2k+1)、β(2k) 的积分公式及 Apostol 型级数。
Relations for Bernoulli--Barnes Numbers and Barnes Zeta Functions
\emph{Barnes $ζ$-function}是为$\Re(x) > 0$和$\Re(z) > n$定义的\[ ζ_n (z, x; \a) := \sum_{\m \in \Z_{\ge 0}^n} \frac{1}{\left(x + m_1 a_1 + \dots + m_n a_n \right)^z} \]，并亚纯地延续到$\C$。专门针对负整数$-k$，Barnes $ζ$ -函数给出了\[ ζ_n (-k, x; \a) = \frac{(-1)^n k!}{(k+n)!} \, B_{k+n} (x; \a) \]，其中$B_k(x; \a)$是一个\emph{Bernoulli--Barnes polynomial}，它也可以通过一个生成函数来定义，该生成函数具有比伯努利多项式稍微更一般的形式。专门化$B_k(0; \a)$得到\emph{Bernoulli--Barnes numbers}。我们展示巴恩斯$ζ$ -功能，伯努利-巴恩斯数和多项式之间的关系，推广各种身份的Agoh，Apostol，Dilcher和欧拉。
We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these established expansions generating sums of the form $\sum_{d: (d,n)=1} f(d)$ (type I) and the Anderson-Apostol sums $\sum_{d|(m,n)} f(d) g(n/d)$ (type II) for any arithmetic functions $f$ and $g$. Our treatment of the type II sums includes a matrix-based factorization method relating the partition function $p(n)$ to arbitrary arithmetic functions $f$. We also conclude…
An analogue of the Rademacher function for generalized Dedekind sums in higher dimension
我们认为广义戴德金和维度$n$，固定$n$ -元组的自然数，定义为周期伯努利函数的值的乘积之和。这包括Zagier和Apostol-Carlitz的广义Dedekind和的高维Dedekind和以及原始Dedekind和。这些都实现为格锥的Todd级数的系数，并从Todd级数的上循环性质满足互易律。利用迭代留数公式，我们计算了定义Dedekind和的格锥的非奇异分解对应的Todd级数的分解的系数。本文给出了一个Laurent多项式，它加到固定指标的广义Dedekind和上，使得它们的分解子有界。我们明确地给出了伯努利数的分母。这推广了由Rademacher函数和经典Dedekind和的差给出的有理函数所起的作用。利用广义Rademacher函数的可积性，将广义Dedekind和与指数和联系起来.我们证明了这个指数和有一个非平凡的界，足以满足Weyl的等分布准则，因此广义Dedekind和的分数部分是等分布的。作为一个例子，对于一个3维的情况和Zagier的高维推广的Dedekind和，我们计算的Laurent多项式相关联。
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