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 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 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 Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 847 ◽  
Author(s):  
Dmitry V. Dolgy ◽  
Dae San Kim ◽  
Jongkyum Kwon ◽  
Taekyun Kim
Keyword(s):  
Random Variables ◽  
Infinite Family ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Integer Power ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Power Sum ◽  
Invariant Integrals

In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on Z p . In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials.

scholarly journals On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 451 ◽  
Author(s):  
Dae Kim ◽  
Taekyun Kim ◽  
Cheon Ryoo ◽  
Yonghong Yao
Keyword(s):  
Integral Representation ◽  
Bernstein Polynomials ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Finite Product ◽  
Bernoulli Numbers And Polynomials ◽  
Special Values ◽  
Invariant Integrals ◽  
Evaluation Problem ◽  
Double Integrals

The q-Bernoulli numbers and polynomials can be given by Witt’s type formulas as p-adic invariant integrals on Z p . We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we study the evaluation problem for the double integrals on Z p of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. This is generalized to the problem of evaluating any finite product of two variable q-Bernstein polynomials. Furthermore, some identities for q-Bernoulli numbers are found.

scholarly journals Extended p-adic q-invariant integrals on Zp associated with applications of umbral calculus

2013 ◽  
Vol 2013 (1) ◽  
pp. 96 ◽  
Author(s):  
Serkan Araci ◽  
Mehmet Acikgoz ◽  
Adem Kilicman
Keyword(s):  
Umbral Calculus ◽  
Invariant Integrals

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 Umbral calculus and special functions

1988 ◽  
Vol 67 (2) ◽  
pp. 174-229 ◽  
Author(s):  
Kazuo Ueno
Keyword(s):  
Special Functions ◽  
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
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