scholarly journals Novel Identities for -Genocchi Numbers and Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Serkan Araci
Keyword(s):  
Bernstein Polynomials ◽  
Bernoulli Numbers ◽  
Genocchi Numbers

The essential aim of this paper is to introduce novel identities forq-Genocchi numbers and polynomials by using the method by T. Kim et al. (article in press). We show that these polynomials are related top-adic analogue of Bernstein polynomials. Also, we derive relations betweenq-Genocchi andq-Bernoulli numbers.

scholarly journals A note on characteristic function for Bernstein polynomials involving special numbers and polynomials

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 543-549
Author(s):  
Buket Simsek
Keyword(s):  
Characteristic Function ◽  
Binomial Distribution ◽  
Bernstein Polynomials ◽  
Random Variable ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Discrete Random Variable ◽  
Euler Numbers ◽  
Negative Order ◽  
Special Numbers And Polynomials

The aim of this present paper is to establish and study generating function associated with a characteristic function for the Bernstein polynomials. By this function, we derive many identities, relations and formulas relevant to moments of discrete random variable for the Bernstein polynomials (binomial distribution), Bernoulli numbers of negative order, Euler numbers of negative order and the Stirling numbers.

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 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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