Nabiullah Khan, Saddam Husain
In the present article, we study Bell based Euler polynomial of order α and investigate some useful correlation formula, summation formula and derivative formula. Also, we introduce some relation of string number of the second kind. Moreover, we drive several important formulae of bell based Euler polynomial by using umbral calculus.
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In this paper we derive two expressions for the Hurwitz zeta function involving the complete Bell polynomials in the restricted case where q is a positive integer greater than 1. The arguments of the complete Bell polynomials comprise the generalised harmonic number functions. These in turn give rise to Euler-Hurwitz series which may then be used to determine identities for combinations of both linear and non-linear Euler sums.
Bell Transforms of Arithmetic Functions: Euler Products, Congruences, and Polynomial Sequences
本文提出一个统一的代数框架，利用形式贝尔变换将算术函数的狄利克雷卷积与无限欧拉型乘积的组合结构联系起来，通过分析指数生成函数的对数导数，建立了贝尔指数与莫比乌斯逆运算之间的显式映射，并应用于推导经典序列的精确消失性质和同余继承性质，包括拉马努金的tau函数和素数着色分区。此外，证明了逆贝尔变换能够无缝恢复经典分区递推关系，并提供了一个离散的组合引擎来生成特殊多项式族，包括经典的Appell和Sheffer序列。
In this paper, we present new explicit simultaneous rational approximations converging sub-exponentially to the values of Bell polynomials at the points of the form $(γ, 1! (2a+1)ζ(2), 2!ζ(3),..., (m-1)!(a+1+(-1)^ma)ζ(m)),$ $m=1,2,...,a,$ $a\in{\mathbb N}.$
Asymptotic for the rightmost zeros of Bell and Eulerian polynomials
本文证明了 Bell 多项式最右负零点的渐近公式，并将方法推广至 Eulerian 多项式等族，得到类似结果。
The relations between the Bernoulli and Eulerian polynomials of higher order and the complete Bell polynomials are found that lead to new identities for the Bernoulli and Eulerian polynomials and numbers of higher order. General form of these identities is considered and generating function for polynomials satisfying this general identity is found.
In this paper, we study degenerate ordered Bell polynomials with the viewpoint of Carlitz's degenerate Bernoulli and Euler polynomials and derive by using umbral calculus some properties and new identities for the degenerate ordered Bell polynomials associated with special polynomials.
In this paper, applying the Faà di Bruno formula and some properties of Bell polynomials, several closed formulas and determinantal expressions involving Stirling numbers of the second kind for higher-order Bernoulli and Euler polynomials are presented.
Recently, several authors have studied the degenerate Bernoulli and Euler polynomials and given some intersting identities of those polynomials. In this paper, we consider the degenerate Bell numbers and polynomials and derive some new identities of those numbers and polynomials associated with special numbers and polynomials. In addition, we investigate some properties of the degenerate Bell polynomials which are derived by using the notion of composita. From our investigation, we give some new relations between the degenerate Bell polynomials and the special polynomials.
Unbounded Oscillation of Euler-Gompertz Diophantine Errors from Bell and Gould Numbers
研究Bell数与Gould数线性组合的丢番图误差δB_n - A_n的渐近行为，证明该误差随n→∞无界振荡，即上下极限分别为正负无穷。
Many works have been done in recent years as to explorations for degenerate versions of some special polynomials and numbers, which began with the introduction of the degenerate Bernoulli and degenerate Euler polynomials by Carlitz. The aim of this paper is to study some properties, recurrence relations and identities related to the degenerate r-Bell polynomials, the two variable degenerate Fubini polynomials and the degenerate r-Stirling numbers of the second kind.
I recent years, studying degenerate versions of some special polynomials, which was initiated by Carlitz in an investigation of the degenerate Bernoulli and Euler polynomials, regained lively interest of mant mathematicains. In this paper, as a degenerate version of polyexponential functions introduced by Hardy, we study degenerate polyexponential functions and derive various properties of them. Also, we introduce new type degenerate Bell polynomials, which are different from the previous studied partially degenerate Bell polynomials and arise naturally in the recent study of degenerate zero-t…
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 Euler-Seidel Matrix Method and Their Applications
本文提出了一种退化欧拉-西德尔矩阵方法，通过引入参数lambda，推导出指数生成函数的退化西德尔公式，并应用于研究新的组合恒等式。
Study on Morgan-Voyce type polynomials with Euler-Seidel algorithm
研究摩根 - 沃伊斯型多项式，通过引入新多项式族建立结构性质，定义变体证明相关和的表达式及推导生成函数，构建欧拉 - 塞德尔矩阵，得到新代数恒等式，扩展矩阵方法在组合分析中的应用。
Some identities involving degenerate Stirling numbers associated with several degenerate polynomials and numbers
本文研究了与退化超调和数及退化Bernoulli、Euler、Bell和Fubini多项式相关的两类退化Stirling数的性质、递推关系和恒等式。
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…
Eisenstein degeneration of Beilinson--Kato classes and circular units
本文研究穿过 Eisenstein 级数临界 $p$-稳定化的 Beilinson–Kato Euler 系统，在多个 $p$-进 $L$-函数消失时建立其与分圆单位的显式分解关系。
A new family of q-Bernstein polynomials: Probabilistic viewpoint
本文提出概率q-伯恩斯坦多项式及其生成函数，利用矩条件建立与斯特林数、伯努利数等的关系，并给出递推、微分性质及p-adic积分表示。
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