A note on poly-Bernoulli polynomials arising from umbral calculus

2013 ◽  
Vol 7 ◽  
pp. 731-744 ◽  
Author(s):  
Dae San Kim ◽  
Taekyun Kim ◽  
Sang-Hun Lee
Keyword(s):  
Bernoulli Polynomials ◽  
Umbral Calculus

scholarly journals Applications of q-Umbral Calculus to Modi…ed Apostol Type q-Bernoulli Polynomials

2017 ◽  
Author(s):  
Mehmet Acikgoz ◽  
Resul Ates ◽  
Ugur Duran ◽  
Serkan Araci
Keyword(s):  
Linear Combination ◽  
Generating Function ◽  
Bernoulli Polynomials ◽  
Umbral Calculus

This article aims to identify the generating function of modi…ed Apostol type q-Bernoulli polynomials. With the aid of this generating function, some properties of modi…ed Apostol type q-Bernoulli polynomials are given. It is shown that aforementioned polynomials are q-Appell. Hence, we make use of these polynomials to have applications on q-Umbral calculus. From those applications, we derive some theorems in order to get Apostol type modi…ed q-Bernoulli polynomials as a linear combination of some known polynomials which we stated in the paper.

scholarly journals Umbral calculus associated with Bernoulli polynomials

2015 ◽  
Vol 147 ◽  
pp. 871-882 ◽  
Author(s):  
Dae San Kim ◽  
Taekyun Kim
Keyword(s):  
Bernoulli Polynomials ◽  
Umbral Calculus

scholarly journals q-Bernoulli polynomials and q-umbral calculus

2014 ◽  
Vol 57 (9) ◽  
pp. 1867-1874 ◽  
Author(s):  
Dae San Kim ◽  
Tae Kyun Kim
Keyword(s):  
Bernoulli Polynomials ◽  
Umbral Calculus

scholarly journals Some Hermite base polynomials on q-umbral algebra

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 961-967 ◽  
Author(s):  
Rahime Dere
Keyword(s):  
Bernoulli Polynomials ◽  
Umbral Calculus ◽  
New Type ◽  
Umbral Algebra

The aim of this paper is to investigate the q-Hermite type polynomials by using umbral calculus methods. Using this method, we derive new type polynomials which are related to the q-Bernoulli polynomials and the q-Hermite type polynomials. Furthermore, we also derive some new identities of those polynomials which are derived from q-umbral calculus.

scholarly journals Degenerate Catalan-Daehee numbers and polynomials of order $ r $ arising from degenerate umbral calculus

2021 ◽  
Vol 7 (3) ◽  
pp. 3845-3865
Author(s):  
Hye Kyung Kim ◽  
◽  
Dmitry V. Dolgy ◽  
Keyword(s):  
Bernoulli Polynomials ◽  
Umbral Calculus ◽  
Bell Polynomials ◽  
Psychological Burden ◽  
Special Polynomials ◽  
Sheffer Sequences ◽  
Special Numbers And Polynomials ◽  
Bernoulli Polynomials And Numbers ◽  
Euler Polynomials And Numbers ◽  
Degenerate Version

<abstract><p>Many mathematicians have studied degenerate versions of some special polynomials and numbers that can take into account the surrounding environment or a person's psychological burden in recent years, and they've discovered some interesting results. Furthermore, one of the most important approaches for finding the combinatorial identities for the degenerate version of special numbers and polynomials is the umbral calculus. The Catalan numbers and the Daehee numbers play important role in connecting relationship between special numbers.</p> <p>In this paper, we first define the degenerate Catalan-Daehee numbers and polynomials and aim to study the relation between well-known special polynomials and degenerate Catalan-Daehee polynomials of order $ r $ as one of the generalizations of the degenerate Catalan-Daehee polynomials by using the degenerate Sheffer sequences. Some of them include the degenerate and other special polynomials and numbers such as the degenerate falling factorials, the degenerate Bernoulli polynomials and numbers of order $ r $, the degenerate Euler polynomials and numbers of order $ r $, the degenerate Daehee polynomials of order $ r $, the degenerate Bell polynomials, and so on.</p></abstract>

scholarly journals Bernoulli polynomials of the second kind and their identities arising from umbral calculus

2016 ◽  
Vol 09 (03) ◽  
pp. 860-869 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Dmitry V. Dolgy ◽  
Jong-Jin Seo
Keyword(s):  
Bernoulli Polynomials ◽  
Umbral Calculus

scholarly journals Bell-Based Bernoulli Polynomials with Applications

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 29
Author(s):  
Ugur Duran ◽  
Serkan Araci ◽  
Mehmet Acikgoz
Keyword(s):  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Umbral Calculus ◽  
Bell Polynomials ◽  
Summation Formulas ◽  
Symmetric Identities

In this paper, we consider Bell-based Stirling polynomials of the second kind and derive some useful relations and properties including some summation formulas related to the Bell polynomials and Stirling numbers of the second kind. Then, we introduce Bell-based Bernoulli polynomials of order α and investigate multifarious correlations and formulas including some summation formulas and derivative properties. Also, we acquire diverse implicit summation formulas and symmetric identities for Bell-based Bernoulli polynomials of order α. Moreover, we attain several interesting formulas of Bell-based Bernoulli polynomials of order α arising from umbral calculus.

scholarly journals Study of Degenerate Poly-Bernoulli Polynomials by λ-Umbral Calculus

2021 ◽  
Vol 129 (1) ◽  
pp. 393-408
Author(s):  
Lee-Chae Jang ◽  
Dae San Kim ◽  
Hanyoung Kim ◽  
Taekyun Kim ◽  
Hyunseok Lee
Keyword(s):  
Bernoulli Polynomials ◽  
Umbral Calculus

scholarly journals Applications of Umbral Calculus Associated withp-Adic Invariant Integrals onZp

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Dae San Kim ◽  
Taekyun Kim
Keyword(s):  
Special Functions ◽  
Bernoulli Polynomials ◽  
Umbral Calculus ◽  
Invariant Integrals ◽  
Umbral Algebra

Recently, Dere and Simsek (2012) have studied the applications of umbral algebra to some special functions. In this paper, we investigate some properties of umbral calculus associated withp-adic invariant integrals onZp. From our properties, we can also derive some interesting identities of Bernoulli 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.

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