A note on degenerate poly-Bernoulli numbers and polynomials
Dae San Kim, Taekyun Kim
本文考虑退化多伯努利多项式，给出了利用退化伯努利多项式和第二类Stirling数计算退化多伯努利多项式的新的显式公式.
In this paper, we consider the degenerate poly-Bernoulli polynomials and present new and explicit formulas for computing them in terms of the degenerate Bernoulli polynomials and Stirling numbers of the second kind.
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In this paper, we study the degenerate version of the new type Euler polynomials, namely degenerate cosine-Euler polynomials and sime-Euler polynomials and also corresponding ones for Bernoulli polynomials, namely degenerate cosine Bernoulli polynomials and degenerate sine-Bernoulli polynomials by considering the degenerate Euler polynomials of complex variable and the degenerate Bernoulli polynomials of complex variable.
Studying degenerate versions of various special polynomials have become an active area of research and yielded many interesting arithmetic and combinatorial results. Here we introduce a degenerate version of polylogarithm function, called the degenerate polylogarithm function. Then we construct new type degenerate Bernoulli polynomials and numbers, called degenerate poly-Bernoulli polynomials and numbers,by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Bernoulli numbers
In this paper, we investigate new class of sequences related to fully degenerate Bernoulli numbers and polynomials. From those sequences, we derive some formulae for the degenerate Bernoulli and Euler polynomials.
The aim of this paper is to study the fully degenerate Bernoulli polynomials and numbers, which are a degenerate version of Bernoulli polynomials and numbers and arise naturally from the Volkenborn integral of the degenerate exponential functions on Zp. We find some explicit expressions for the fully degenerate Bernoulli polynomials and numbers in terms of the degenerate Stirling numbers of the second kind, the degenerate r-Stirling numbers of the second kind and of the degenerate Stirling polynomials. We also consider the degenerate poly-Bernoulli polynomials and derive explicit representatio…
On degenerate Carlitz q-Bernoulli polynomials
本文考虑退化Carlitz q-Bernoulli数和多项式，研究了退化Carlitz q-Bernoulli多项式的一些性质。
Fully degenerate poly-Bernoulli numbers and polynomials
本文引入了新的完全退化的poly-Bernoulli数和多项式，并研究了它们的一些性质。利用这些性质，我们得到了完全退化的多伯努利数和多项式的一些恒等式。
In this paper, we consider the degenerate multi-poly-Bernoulli numbers and polynomials which are defined by means of the multiple polylogarithms and degenerate versions of the multi-poly-Bernoulli numbers and polynomials. We investigate some properties for those numbers and polynomials. In addition, we give some identities and relations for the degenerate multi-poly-Bernoulli numbers and polynomials.
As is well-known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate version of such functions and polynomials, degenerate polylogarithm functions were introduced and degenertae poly-Bernoulli polynomials were defined by means of the degenerate polylogarithm functions, and some their properties were investigated. The aim of this paper is to furthur study some properties of the degenertae poly-Bernoulli polynomials by using three formulas coming from the recently developed lambda-umbral calculus. In more detail, among other things, we represent t…
In this paper, we consider the problem of representing any polynomial in terms of the degenerate Bernoulli polynomials and more generally of the higher-order degenerate Bernoulli polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.
The purpose of this paper is to give symmetric identities for higher-order degenerate q- Bernoulli polynomials arising from the p-adic q-integral on Zp.
Some identities of Carlitz degenerate Bernoulli numbers and polynomials
本文研究了Carlitz退化Bernoulli数和多项式，给出了与之相关的一些公式和恒等式。
We introduce the degenerate Bernoulli numbers of the second kind as a degenerate version of the Bernoulli numbers of the second kind. We derive a family of nonlinear differential equations satisfied by a function closely related to the generating function for those numbers. We obtain explicit expressions for the coefficients appearing in those differential equations and the degenerate Bernoulli numbers of the second kind. In addition, as an application and from those differential equations we have an identity expressing the degenerate Bernoulli numbers of the second kind in terms of those numb…
In a previous paper, Rahmani introduced a new family of p-Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. Motivated by this paper and as a degenerate version of those numbers and polynomials, we introduce the generalized degenerate Bernoulli numbers and polynomials again by using the Gauss hypergeometric function. In addition, we introduce the degenerate type Eulerian numbers as a degenerate version of Eulerian numbers. For the generalized degenerate Bernoulli numbers, we express them in terms of the degenerate Stirling numbers of the second kind, of the degene…
Some applications of degenerate poly-Bernoulli numbers and polynomials
本文考虑Zp上退化的多伯努利数和多项式与多对数函数和p-adic不变积分的关系。利用Umbral演算，我们得到了这些数和多项式的一些恒等式
I recent years, many mathematicians studied various degenerate version of some spcial polynomials of which quite a few interesting results were discovered. In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and study some identities and expressions for these polynomials. Specially, we obtain a relation between the type 2 degenerate Bernoulli polynomials of the second kind and degenerate Bernoulli polynomials of the second kind, and identity involving hihger-order analogues of those polynomials and the degenerate stirling…
Recently, Masjed-Jamei-Beyki-Koepf studied the so called new type Euler polynomials without making use of Euler polynomials of complex variable. Here we study degenerate and type 2 versions of these new type Euler polynomials, namely the type 2 degenerate cosine-Euler and type 2 degenerate sine-Euler polynomials and also the corresponding ones for Bernoulli polynomials, namely the type 2 degenerate cosine-Bernoulli and type 2 degenerate sine-Bernoulli polynomials by considering the degenerate Euler and degenerate Bernoulli polynomials of complex variable and by treating the real and imaginary …
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.
Degenerate Algorithms for degenerate Bernoulli and Euler numbers
本文提出退化A和B算法，通过引入参数lambda推导出退化伯努利和欧拉数的显式公式，并建立生成函数之间的关系。
In this work we study the properties of a new algebraic variant of the degenerate Bernoulli polynomial $\tildeβ_{k}(m,x)$ and study the corresponding degenerate Bernoulli number $\tildeβ_{k}(m,1)=m^{k}β_{k}(1/m)$, where $β_{k}(λ), λ\neq 0$ is the standard degenerate Bernoulli number. Our approach relies on a new algebraic framework for generating functions and the action of a symbolic evaluation function on powers of polynomials. We show that $\tildeβ_{k}(m,x)$ displays surprising links with other mathematical objects (such as Circulant matrices and Galois fields) and enjoys many interesting a…
A study of degenerate Bernoulli and Euler numbers via operators
本文引入新型算子类，推导退化欧拉数与2型退化欧拉数的显式表达式，建立退化伯努利数与退化欧拉数的关系，为特殊数研究提供新方法。
Recently, Kim-Kim Introduced some interesting identities of symmetry for q-Bernoulli polynomials under symmetry group of degree n. In this paper, we study the degenerate q-Euler polynomials and derive some identities of symmetry for these polynomials arising from the p-adic q-integral on Zp.
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。
In recent years, studying degenerate versions of various special polynomials and numbers have attracted many mathematicians. Here we introduce degenerate type 2 Bernoulli polynomials, fully degenerate type 2 Bernoulli polynomials and degenerate type 2 Euler polynomials, and their corresponding numbers, as degenerate and type 2 versions of Bernoulli and Euler numbers. Regarding to those polynomials and numbers, we derive some identities, distribution relations, Witt type formulas and analogues for the Bernoulli's interpretation of powers of the first $m$ positive integers in terms of Bernoulli …
Degenerate Sheffer-type polynomials and degenerate Sheffer polynomials associated with a random variable
本文研究退化Sheffer型多项式并引入与随机变量Y相关联的退化Sheffer多项式，推导其性质，同时给出高阶退化Bernoulli与Euler多项式的新结果。
Representations by probabilistic Bernoulli and degenerate Bernoulli polynomials
本文利用概率伯努利和退化伯努利多项式表示任意多项式，借助阴影演算给出显式公式，并针对离散和连续随机变量展示结果。
Here we consider the degenerate Bernstein polynomials as a degenerate version of Bernstein polynomials, which are motivated by Simsek's recent work 'Generating functions for unification of the multidimensional Bernstein polynomials and their applications'([15,16]) and Carlitz's degenerate Bernoulli polynomials. We derived thier generating function, symmetric identities, recurrence relations, and some connections with generalized falling factorial polynomials, higher-order degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind.
Higher-order Bernoulli and poly-Bernoulli mixed type polynomials
本文考虑高阶Bernoulli和poly-Bernoulli混合型多项式，并给出了这类多项式的一些有趣的恒等式。
Barnes' multiple Bernoulli and poly-Bernoulli mixed-type polynomials
本文考虑Barnes多重Bernoulli和poly-Bernoulli混合型多项式。从这类多项式的Sheffer序列的性质出发，我们得到了新的有趣的恒等式。
We review and discuss some results on the representation of Bernoulli, poly-Bernoulli numbers, and Bernoulli and Cauchy polynomials in terms of Stirling numbers of the first or second kind, or in terms of r-Stirling numbers.
Relations between Multi-Poly-Bernoulli numbers and Poly-Bernoulli numbers of negative index
Poly-Bernoulli数$B_n^{(k)}\in\mathbb{Q}$ \，（$n \geq 0$，\，$k \in \mathbb{Z}$）由Kaneko在1997年定义。多重多伯努利数\，$B_n^{(k_1,k_2,\ldots, k_r)}$，是Kaneko的多伯努利数\，$B_n^{(k)}$的代。重点研究了多重多伯努利数与负指数多伯努利数之间的关系。在第二节中，我们引入了Kamano证明的负指数多重Poly Bernoulli数的一个恒等式。在第三节中，作为主要结果，我们特别介绍了多重Poly-Bernoulli数与负指数Poly-Bernoulli数之间的一些关系。
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。
The multi-poly-Bernoulli numbers are generalizations of the Bernoulli numbers. In this paper, we will prove Kummer-type congruences for multi-poly-Bernoulli numbers via $p$-adic distributions.
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.
We present a relationship between the generalized hyperharmonic numbers and the poly-Bernoulli polynomials, motivated from the connections between harmonic and Bernoulli numbers. This relationship yields numerous identities for the hyper-sums and several congruences.
Hermite and poly-Bernoulli mixed-type polynomials
本文考虑Hermite和poly-Bernoulli混合型多项式，并研究了这类多项式的性质。最后，给出了高阶Stirling数、Bernoulli数和Frobenius-欧拉多项式的各种恒等式.
On Some Explicit Formulas for Bernoulli Numbers and Polynomials
给出了Bernoulli数和伯努利多项式的几种显式的直接初等证明。作为我们的证明方法的副产品，我们提供了自然的定义广义伯努利数和多项式的复杂顺序。
Some identities for Bernoulli polynomials involving Chebyshev polynomials
本文导出了与切比雪夫多项式有关的Bernoulli、Euler和Hermite多项式的一些新的有趣的恒等式。
A note on poly-Bernoulli numbers and polynomials of the second kind
本文研究了第二类多伯努利数和多项式，给出了计算第二类多伯努利数和第二类Stirling数的新的显式公式.
Poly-Bernoulli polynomials arising from umbral calculus
本文给出了由投影演算导出的多Bernoulli数和多项式的一些递推公式和新的有趣的恒等式。
Higher-order frobenius-Euler and poly-Bernoulli mixed type polynomials
本文考虑由多对数函数导出的与多伯努利多项式相联系的高阶Frobenius-Euler多项式。这些多项式被称为高阶Frobenius-Euler和poly-Bernoulli混合型多项式。本文的目的是给出由投影演算产生的多项式的各种恒等式。
We realize that geometric polynomials and p-Bernoulli polynomials and numbers are closely related with an integral representation. Therefore, using geometric polynomials, we extend some properties of Bernoulli polynomials and numbers such as recurrence relations, telescopic formula and Raabe's formula to p-Bernoulli polynomials and numbers. In particular cases of these results, we establish some new results for Bernoulli polynomials and numbers. Moreover, we evaluate a Faulhaber-type summation in terms of p-Bernoulli polynomials.
On the weighted q-Bernoulli numbers and polynomials
本文考虑了与Carlitz的q-Bernoulli数和多项式不同类型的加权q-Bernoulli数和多项式。从这些数和多项式出发，我们得到了一些有趣的公式和恒等式。
q-Bernstein polynomials, q-Stirling numbers and q-Bernoulli polynomials
本文给出了涉及Phillips q-伯恩斯坦多项式的新恒等式，并得到了与q-Stirling数和q-伯努利多项式有关的q-Bernstein多项式的一些有趣的性质.
q-Bernoulli polynomials and q-umbral calculus
本文研究了由q-影演算导出的q-伯努利多项式的一些性质。最后，我们得到了q-伯努利多项式的一些有趣的恒等式。
Generalizations of Poly-Bernoulli numbers and polynomials
本文将多伯努利数$B_n^{(k)}$、多伯努利多项式$B_n^{k}{(t)}$和广义多伯努利数$B_{n}^{(k)}(a,b)$的概念推广到$B_{n}^{(k)}(t,a,b,c)$，称之为依赖于实参数的广义多伯努利多项式\textit{a,b,c} .建立了这些多项式的一些性质以及$B_n^{k}$、$B_n^{(k)}(t)$、$B_{n}^{(k)}(a,b)$和$B_{n}^{(k)}(t,a,b,c)$之间的关系
New Convolution Identities for Hypergeometric Bernoulli Polynomials
给出了超几何伯努利多项式的新卷积恒等式。讨论了证明这些恒等式的两种不同方法，对应于超几何伯努利多项式作为Appell序列的两种等价定义。
We introduce multi-poly-Bernoulli-Carlitz numbers, function field analogues of multi-poly-Bernoulli numbers of Imatomi-Kaneko-Takeda. We explicitly describe multi-poly-Bernoulli Carlitz numbers in terms of the Carlitz factorial and the Stirling-Carlitz numbers of the second kind and also show their relationships with function field analogues of finite multiple zeta values.
In this paper, we establish an identity for Bernoulli's generalized polynomials. We deduce generalizations for many relations involving classical Bernoulli numbers or polynomials. In particular, we generalize a recent Gessel identity.
A note on q-Bernoulli numbers and polynomials
利用p-adic q-积分，研究了q-Bernoulli数和高阶多项式。
Identities involving values of Bernstein, q-Bernoulli and q-Euler polynomials
本文给出了q-Bernoulli多项式、q-Euler多项式和伯恩斯坦多项式的值之间的关系。利用这些关系，我们得到了q-Bernoulli多项式、q-Euler多项式和伯恩斯坦多项式的一些有趣的恒等式.
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.
We continue our study on relationships between Bernoulli polynomials and balancing (Lucas-balancing) polynomials. From these polynomial relations, we deduce new combinatorial identities with Fibonacci (Lucas) and Bernoulli numbers. Moreover, we prove a special identity involving Bernoulli polynomials and Fibonacci numbers in arithmetic progression. Special cases and some corollaries will highlight interesting aspects of our findings. Our results complement and generalize these of Frontczak (2019).
On the (h,q)-zeta functions associated with (h,q)-Bernoulli numbers and polynomials
本文研究了与（h，q）-Bernoulli数和多项式相关的（h，q）-zeta函数.
On the twisted $q$-zeta functions and $q$-Bernoulli polynomials
研究了扭q zeta函数和扭q伯努利多项式
As properties of poly-Bernoulli numbers, a number of formulas such as the duality formula, explicit formula using the Stirling numbers of the second kind and periodicity for negative upper-index have been established. For the multi-indexed poly-Bernoulli numbers generalized by Kaneko-Tsumura, among such properties only the duality formula was obtained. In this paper, we restrict the double-indexed poly-Bernoulli numbers and show the explicit formula using the Stirling numbers of the second kind and periodicity for negative upper-index for them. Further, we define the variant of multiple-indexe…
Identities of symmetry for higher-order q-Bernoulli polynomials
最近，Kim将高阶Carlitz的q-伯努利多项式表示为Zp上的q-Volkenborn积分.文[13]中提出了一个问题，即寻找与Carlitz q-伯努利多项式有关的伯努利多项式对称性的推广公式。本文给出了由Zp上的多元q-Volkenborn积分导出的高阶Carlitz q-伯努利多项式的几个新的对称恒等式。我们注意到，它们只是对这个问题的部分回答。.
We present a symbolic representation for the poly-Bernoulli numbers. This allows us to prove several new iterated integral representations for the poly-Bernoulli numbers, including an integral transform of the Bernoulli-Barnes numbers. We also deduce some new recurrences for the poly-Bernoulli numbers. Finally, we use these results to present a new iterated integral representation for the Arakawa-Kaneko zeta function, including a nonlinear integral transform of the Barnes zeta function.
We show that certain weighted Fibonacci and Lucas series can always be expressed as linear combinations of polylogarithms. In some special cases we evaluate the series in terms of Bernoulli polynomials, making use of the connection between these polynomials and the polylogarithm.
We prove a duality formula for certain sums of values of poly-Bernoulli polynomials which generalizes dualities for poly-Bernoulli numbers. We first compute two types of generating functions for these sums, from which the duality formula is apparent. Secondly we give an analytic proof of the duality from the viewpoint of our previous study of zeta-functions of Arakawa-Kaneko type. As an application, we give a formula that relates poly-Bernoulli numbers to the Genocchi numbers.
On hypergeometric Bernoulli numbers and polynomials
本文给出了超几何Bernoulli数和多项式的若干性质，包括乘积和恒等式、微分方程和递推公式。
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