scholarly journals Umbral Calculus and the Frobenius-Euler Polynomials

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Dae San Kim ◽  
Taekyun Kim ◽  
Sang-Hun Lee
Keyword(s):  
Umbral Calculus ◽  
Euler Polynomials

We study some properties of umbral calculus related to the Appell sequence. From those properties, we derive new and interesting identities of the Frobenius-Euler polynomials.

scholarly journals Symmetric identity for polynomial sequences satisfying A′ n +1(x) = (n + 1)An (x)

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Farid Bencherif ◽  
Rachid Boumahdi ◽  
Tarek Garici
Keyword(s):  
Bernoulli Polynomials ◽  
Dirichlet Character ◽  
Umbral Calculus ◽  
Euler Polynomials ◽  
Polynomial Sequences ◽  
Symmetric Identity ◽  
Primitive Dirichlet Character ◽  
Generalized Bernoulli Polynomials

Abstract Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying A′ n +1(x) = (n + 1)An (x) with A 0(x) a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, Apostol-Euler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.

scholarly journals On the type 2 poly-Bernoulli polynomials associated with umbral calculus

2021 ◽  
Vol 19 (1) ◽  
pp. 878-887
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Dmitry V. Dolgy ◽  
Jin-Woo Park
Keyword(s):  
Bernoulli Polynomials ◽  
Higher Order ◽  
Umbral Calculus ◽  
Euler Polynomials ◽  
Special Polynomials

Abstract Type 2 poly-Bernoulli polynomials were introduced recently with the help of modified polyexponential functions. In this paper, we investigate several properties and identities associated with those polynomials arising from umbral calculus techniques. In particular, we express the type 2 poly-Bernoulli polynomials in terms of several special polynomials, like higher-order Cauchy polynomials, higher-order Euler polynomials, and higher-order Frobenius-Euler polynomials.

scholarly journals Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 47 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Bernstein Polynomials ◽  
Umbral Calculus ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Special Numbers And Polynomials ◽  
Degenerate Bernstein Polynomials ◽  
Combinatorial Methods

In recent years, intensive studies on degenerate versions of various special numbers and polynomials have been done by means of generating functions, combinatorial methods, umbral calculus, p-adic analysis and differential equations. The degenerate Bernstein polynomials and operators were recently introduced as degenerate versions of the classical Bernstein polynomials and operators. Herein, we firstly derive some of their basic properties. Secondly, we explore some properties of the degenerate Euler numbers and polynomials and also their relations with the degenerate Bernstein polynomials.

scholarly journals Frobenius-Euler polynomials and umbral calculus in the p-adic case

2012 ◽  
Vol 2012 (1) ◽  
Author(s):  
Dae San Kim ◽  
Taekyun Kim ◽  
Sang-Hun Lee ◽  
Seog-Hoon Rim
Keyword(s):  
Umbral Calculus ◽  
Euler Polynomials ◽  
Adic Case

scholarly journals Apostol-Euler polynomials arising from umbral calculus

2013 ◽  
Vol 2013 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Toufik Mansour ◽  
Seog-Hoon Rim ◽  
Sang-Hun Lee
Keyword(s):  
Umbral Calculus ◽  
Euler Polynomials

scholarly journals Some recurrence formulas for the q-Bernoulli and q-Euler polynomials

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 663-669
Author(s):  
Paçin Dere
Keyword(s):  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Umbral Calculus ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Derivative Operator ◽  
Important Place ◽  
Quantum Calculus ◽  
Recurrence Formulas ◽  
Special Polynomials

The recurrence relations have a very important place for the special polynomials such as q-Appell polynomials. In this paper, we give some recurrence formulas that allow us a better understanding of q-Appell polynomials. We investigate the q-Bernoulli polynomials and q-Euler polynomials, which are q-Appell polynomials, and we obtain their recurrence formulas by using the methods of the q-umbral calculus and the quantum calculus. Our methods include some operators which are quite handy for obtaining relations for the q-Appell polynomials. Especially, some applications of q-derivative operator are used in this work.

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.

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