Explicit Formulas for Special Values of the Bell Polynomials of the Second Kind and for the Euler Numbers and Polynomials

2017 ◽  
Vol 14 (3) ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Bell Polynomials ◽  
Euler Numbers ◽  
Explicit Formulas ◽  
Special Values ◽  
Euler Numbers And Polynomials

scholarly journals Explicit Formulas Involving -Euler Numbers and Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Serkan Araci ◽  
Mehmet Acikgoz ◽  
Jong Jin Seo
Keyword(s):  
Generating Function ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Explicit Formulas ◽  
Derivative Operator ◽  
Euler Numbers And Polynomials ◽  
Integer Ring

We deal with -Euler numbers and -Bernoulli numbers. We derive some interesting relations for -Euler numbers and polynomials by using their generating function and derivative operator. Also, we derive relations between the -Euler numbers and -Bernoulli numbers via the -adic -integral in the -adic integer ring.

scholarly journals Some Symmetry Identities for Carlitz’s Type Degenerate Twisted (p,q)-Euler Polynomials Related to Alternating Twisted (p,q)-Sums

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1371
Author(s):  
Cheon-Seoung Ryoo
Keyword(s):  
Euler Polynomials ◽  
Euler Numbers ◽  
Explicit Formulas ◽  
Euler Numbers And Polynomials ◽  
New Form ◽  
Symmetric Properties

In this paper, we define a new form of Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials by generalizing the degenerate Euler numbers and polynomials, Carlitz’s type degenerate q-Euler numbers and polynomials. Some interesting identities, explicit formulas, symmetric properties, a connection with Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials are obtained. Finally, we investigate the zeros of the Carlitz’s type degenerate twisted (p,q)-Euler polynomials by using computer.

scholarly journals Closed formulas for special bell polynomials by Stirling numbers and associate Stirling numbers

2020 ◽  
Vol 108 (122) ◽  
pp. 131-136
Author(s):  
Feng Qi ◽  
Dongkyu Lim
Keyword(s):  
Explicit Formula ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Special Values

We derive two explicit formulas for two sequences of special values of the Bell polynomials of the second kind in terms of associate Stirling numbers of the second kind, give an explicit formula for associate Stirling numbers of the second kind in terms of the Stirling numbers of the second kind, and, consequently, present two explicit formulas for two sequences of special values of the Bell polynomials of the second kind in terms of the Stirling numbers of the second kind.

scholarly journals Degenerate Bell polynomials associated with umbral calculus

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Han-Young Kim ◽  
Hyunseok Lee ◽  
Lee-Chae Jang
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Umbral Calculus ◽  
Bell Polynomials ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Type Formula ◽  
Euler Numbers And Polynomials ◽  
Special Numbers And Polynomials

Abstract Carlitz initiated a study of degenerate Bernoulli and Euler numbers and polynomials which is the pioneering work on degenerate versions of special numbers and polynomials. In recent years, studying degenerate versions regained lively interest of some mathematicians. The purpose of this paper is to study degenerate Bell polynomials by using umbral calculus and generating functions. We derive several properties of the degenerate Bell polynomials including recurrence relations, Dobinski-type formula, and derivatives. In addition, we represent various known families of polynomials such as Euler polynomials, modified degenerate poly-Bernoulli polynomials, degenerate Bernoulli polynomials of the second kind, and falling factorials in terms of degenerate Bell polynomials and vice versa.

On a generalization of the Rényi–Srivastava characterization of the Poisson law

2021 ◽  
Vol 58 (1) ◽  
pp. 68-82
Author(s):  
Jean-Renaud Pycke
Keyword(s):  
Life Insurance ◽  
Collective Model ◽  
New Method ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Compound Distributions ◽  
Poisson Law ◽  
Pierre Loti

AbstractWe give a new method of proof for a result of D. Pierre-Loti-Viaud and P. Boulongne which can be seen as a generalization of a characterization of Poisson law due to Rényi and Srivastava. We also provide explicit formulas, in terms of Bell polynomials, for the moments of the compound distributions occurring in the extended collective model in non-life insurance.

scholarly journals On p-Adic Fermionic Integrals of q-Bernstein Polynomials Associated with q-Euler Numbers and Polynomials †

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 311 ◽  
Author(s):  
Lee-Chae Jang ◽  
Taekyun Kim ◽  
Dae Kim ◽  
Dmitry Dolgy
Keyword(s):  
Bernstein Polynomials ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials

We study a q-analogue of Euler numbers and polynomials naturally arising from the p-adic fermionic integrals on Zp and investigate some properties for these numbers and polynomials. Then we will consider p-adic fermionic integrals on Zp of the two variable q-Bernstein polynomials, recently introduced by Kim, and demonstrate that they can be written in terms of the q-analogues of Euler numbers. Further, from such p-adic integrals we will derive some identities for the q-analogues of Euler numbers.

scholarly journals Some Symmetric Identities for the Multiple (p, q)-Hurwitz-Euler eta Function

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 645 ◽  
Author(s):  
Kyung-Won Hwang ◽  
Cheon Seoung Ryoo
Keyword(s):  
Higher Order ◽  
Complex Field ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Symmetric Identities ◽  
Symmetric Properties

The main purpose of this paper is to find some interesting symmetric identities for the ( p , q ) -Hurwitz-Euler eta function in a complex field. Firstly, we define the multiple ( p , q ) -Hurwitz-Euler eta function by generalizing the Carlitz’s form ( p , q ) -Euler numbers and polynomials. We find some formulas and properties involved in Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order. We find new symmetric identities for multiple ( p , q ) -Hurwitz-Euler eta functions. We also obtain symmetric identities for Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order by using symmetry about multiple ( p , q ) -Hurwitz-Euler eta functions. Finally, we study the distribution and symmetric properties of the zero of Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order.

scholarly journals Special values of the Bell polynomials of the second kind for some sequences and functions

2020 ◽  
Vol 491 (2) ◽  
pp. 124382
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Dongkyu Lim ◽  
Yong-Hong Yao
Keyword(s):  
Bell Polynomials ◽  
Special Values
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