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Open AccessArticle

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

1
Department of Mathematics, Sogang University, Seoul 121-742, Korea
2
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
3
Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea
4
Department of Mathematics, Hannam University, Daejeon 306-791, Korea
5
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China
*
Author to whom correspondence should be addressed.
Symmetry 2018, 10(10), 451; https://doi.org/10.3390/sym10100451
Received: 28 July 2018 / Revised: 19 September 2018 / Accepted: 20 September 2018 / Published: 1 October 2018
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with their Applications)
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. View Full-Text
Keywords: q-Bernoulli numbers; q-Bernoulli polynomials; two variable q-Bernstein polynomials; two variable q-Bernstein operators; p-adic integral on ℤp q-Bernoulli numbers; q-Bernoulli polynomials; two variable q-Bernstein polynomials; two variable q-Bernstein operators; p-adic integral on ℤp
MDPI and ACS Style

Kim, D.S.; Kim, T.; Ryoo, C.S.; Yao, Y. On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials. Symmetry 2018, 10, 451.

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