scholarly journals Some Identities on theq-Bernoulli Numbers and Polynomials with Weight 0

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
T. Kim ◽  
J. Choi ◽  
Y. H. Kim
Keyword(s):  
Integral Representation ◽  
Bernstein Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials

Recently, Kim (2011) has introduced theq-Bernoulli numbers with weightα. In this paper, we consider theq-Bernoulli numbers and polynomials with weightα=0and givep-adicq-integral representation of Bernstein polynomials associated withq-Bernoulli numbers and polynomials with weight0. From these integral representation onℤp, we derive some interesting identities on theq-Bernoulli numbers and polynomials with weight0.

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 On p-Adic Integral Representation of q-Bernoulli Numbers Arising from two Variable q-Bernstein Polynomials

2018 ◽  
Author(s):  
C.S. Ryoo ◽  
T. Kim ◽  
D.S. Kim ◽  
Y. Yao
Keyword(s):  
Integral Representation ◽  
Bernstein Polynomials ◽  
Bernoulli Numbers

In this paper, we study the p-adic integral representation on Zp of q-Bernoulli numbers arising from two variable q-Bernstein polynomials and investigate some properties for the q-Bernoulli numbers. In addition, we give some new identities of q-Bernoulli numbers.

scholarly journals On (q, r, w)-Stirling Numbers of the Second Kind

2017 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Linear Combination ◽  
Generating Function ◽  
Bernstein Polynomials ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Exponential Generating Function

In this paper, we introduce a new generalization of the r-Stirling numbers of the second kind based on the q-numbers via an exponential generating function. We investigate their some properties and derive several relations among q-Bernoulli numbers and polynomials, and newly de…ned (q, r, w)-Stirling numbers of the second kind. We also obtain q-Bernstein polynomials as a linear combination of (q, r, w)-Stirling numbers of the second kind and q-Bernoulli polynomials in w.

scholarly journals On -Adic Analogue of -Bernstein Polynomials and Related Integrals

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
T. Kim ◽  
J. Choi ◽  
Y. H. Kim ◽  
L. C. Jang
Keyword(s):  
Bernstein Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials

Recently, Kim's work (in press) introduced -Bernstein polynomials which are different Phillips' -Bernstein polynomials introduced in the work by (Phillips, 1996; 1997). The purpose of this paper is to study some properties of several type Kim's -Bernstein polynomials to express the -adic -integral of these polynomials on associated with Carlitz's -Bernoulli numbers and polynomials. Finally, we also derive some relations on the -adic -integral of the products of several type Kim's -Bernstein polynomials and the powers of them on .

scholarly journals Some Identities of Bernoulli Numbers and Polynomials Associated with Bernstein Polynomials

2010 ◽  
Vol 2010 ◽  
pp. 1-7 ◽  
Author(s):  
Min-Soo Kim ◽  
Taekyun Kim ◽  
Byungje Lee ◽  
Cheon-Seoung Ryoo
Keyword(s):  
Bernstein Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials

scholarly journals Some Identities of Bernoulli Numbers and Polynomials Associated with Bernstein Polynomials

2010 ◽  
Vol 2010 (1) ◽  
pp. 305018 ◽  
Author(s):  
Min-Soo Kim ◽  
Taekyun Kim ◽  
Byungje Lee ◽  
Cheon-Seoung Ryoo
Keyword(s):  
Bernstein Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials

scholarly journals -Bernstein Polynomials Associated with -Stirling Numbers and Carlitz's -Bernoulli Numbers

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
T. Kim ◽  
J. Choi ◽  
Y. H. Kim
Keyword(s):  
Integral Representation ◽  
Binomial Distribution ◽  
Bernstein Polynomials ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Bernstein Type

Recently, Kim (2011) introduced -Bernstein polynomials which are different -Bernstein polynomials of Phillips (1997). In this paper, we give a -adic -integral representation for -Bernstein type polynomials and investigate some interesting identities of -Bernstein type polynomials associated with -extensions of the binomial distribution, -Stirling numbers, and Carlitz's -Bernoulli numbers.

scholarly journals Some results on degenerate Daehee and Bernoulli numbers and polynomials

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim ◽  
Jongkyum Kwon
Keyword(s):  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials

scholarly journals New type of degenerate Daehee polynomials of the second kind

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sunil Kumar Sharma ◽  
Waseem A. Khan ◽  
Serkan Araci ◽  
Sameh S. Ahmed
Keyword(s):  
Generating Function ◽  
Special Class ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Bernoulli Numbers And Polynomials ◽  
New Type ◽  
Order Degenerate ◽  
Math Phys

Abstract Recently, Kim and Kim (Russ. J. Math. Phys. 27(2):227–235, 2020) have studied new type degenerate Bernoulli numbers and polynomials by making use of degenerate logarithm. Motivated by (Kim and Kim in Russ. J. Math. Phys. 27(2):227–235, 2020), we consider a special class of polynomials, which we call a new type of degenerate Daehee numbers and polynomials of the second kind. By using their generating function, we derive some new relations including the degenerate Stirling numbers of the first and second kinds. Moreover, we introduce a new type of higher-order degenerate Daehee polynomials of the second kind. We also derive some new identities and properties of this type of polynomials.

Sign in / Sign up

Export Citation Format

Share Document