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 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 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 (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 Some Identities of Symmetry for the Generalized Bernoulli Numbers and Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Taekyun Kim ◽  
Seog-Hoon Rim ◽  
Byungje Lee
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Invariant Integral ◽  
Generalized Bernoulli Polynomials ◽  
Symmetric Properties

By the properties ofp-adic invariant integral onℤp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties ofp-adic invariant integral onℤp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

scholarly journals On the Symmetries of theq-Bernoulli Polynomials

2008 ◽  
Vol 2008 ◽  
pp. 1-7 ◽  
Author(s):  
Taekyun Kim
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Reflection Symmetry ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sum ◽  
Congruence Properties

Kupershmidt and Tuenter have introduced reflection symmetries for theq-Bernoulli numbers and the Bernoulli polynomials in (2005), (2001), respectively. However, they have not dealt with congruence properties for these numbers entirely. Kupershmidt gave a quantization of the reflection symmetry for the classical Bernoulli polynomials. Tuenter derived a symmetry of power sum polynomials and the classical Bernoulli numbers. In this paper, we study the new symmetries of theq-Bernoulli numbers and polynomials, which are different from Kupershmidt's and Tuenter's results. By using our symmetries for theq-Bernoulli polynomials, we can obtain some interesting relationships betweenq-Bernoulli numbers and polynomials.

scholarly journals Some identities relating to degenerate Bernoulli polynomials

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 905-912 ◽  
Author(s):  
Taekyun Kim ◽  
Dae Kim ◽  
Hyuck-In Kwon
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials

Recently, Carlitz degenerate Bernoulli numbers and polynomials have been studied by several authors (see [3,4]). In this paper, we consider new degenerate Bernoulli numbers and polynomials, different from Carlitz degenerate Bernoulli numbers and polynomials, and give some formulae and identities related to these numbers and polynomials.

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 -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 A symbolic approach to some identities for Bernoulli–Barnes polynomials

2016 ◽  
Vol 12 (03) ◽  
pp. 649-662 ◽  
Author(s):  
Lin Jiu ◽  
Victor H. Moll ◽  
Christophe Vignat
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Symbolic Calculus ◽  
Specific Expression ◽  
Bernoulli Numbers And Polynomials ◽  
Self Duality ◽  
New Family ◽  
Symbolic Approach ◽  
Dual Sequences

The Bernoulli–Barnes polynomials are defined as a natural multidimensional extension of the classical Bernoulli polynomials. Many of the properties of the Bernoulli polynomials admit extensions to this new family. A specific expression involving the Bernoulli–Barnes polynomials has recently appeared in the context of self-dual sequences. The work presented here provides a proof of this self-duality using the symbolic calculus attached to Bernoulli numbers and polynomials. Several properties of the Bernoulli–Barnes polynomials are established by this procedure.

scholarly journals Some Identities between the Extended -Bernstein Polynomials with Weight and -Bernoulli Polynomials with Weight (,)

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
H. Y. Lee ◽  
C. S. Ryoo
Keyword(s):  
Bernstein Polynomials ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers

Using bosonic -adic -integral on , we give some interesting relationships between -Bernoulli numbers with weight (,) and -Bernstein polynomials with weight . Also, using -Bernstein polynomials with two variables, we derive some interesting properties associated with -Bernoulli numbers with weight (,).

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