Some identities of Bernoulli numbers and polynomials associated with Bernstein polynomials. (English) Zbl 1221.11059

The authors introduce the Bernstein polynomials on the ring of \(p\)-adic integers \(\mathbb Z_p\) and investigate certain properties of the Bernstein polynomials related to the bosonic \(p\)-adic integrals on \(\mathbb Z_p\).


11B68 Bernoulli and Euler numbers and polynomials
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
Full Text: DOI arXiv EuDML


[1] Acikgoz M, Araci S: A study on the integral of the product of several type Bernstein polynomials.IST Transaction of Applied Mathematics-Modelling and Simulation. In press
[2] Acikgoz, M; Araci, S, On the generating function of the Bernstein polynomials, (2010)
[3] Simsek, Y; Acikgoz, M, A new generating function of ([inlineequation not available: see fulltext.]-) Bernstein-type polynomials and their interpolation function, No. 2010, 12, (2010)
[4] Bernstein, S, Demonstration du theoreme de Weierstrass, fondee sur le calcul des probabilities, Communications of the Kharkov Mathematical Society, 13, 1-2, (1913)
[5] Jang, L-C; Kim, W-J; Simsek, Y, A study on the p-adic integral representation on [inlineequation not available: see fulltext.] associated with Bernstein and Bernoulli polynomials, No. 2010, 6, (2010)
[6] Kim, T; Jang, L-C; Yi, H, A note on the modified [inlineequation not available: see fulltext.]-Bernstein polynomials, No. 2010, 12, (2010)
[7] Phillips, GM, Bernstein polynomials based on the [inlineequation not available: see fulltext.]-integers, Annals of Numerical Mathematics, 4, 511-518, (1997) · Zbl 0881.41008
[8] Kim, T, On a [inlineequation not available: see fulltext.]-analogue of the [inlineequation not available: see fulltext.]-adic log gamma functions and related integrals, Journal of Number Theory, 76, 320-329, (1999) · Zbl 0941.11048
[9] Kim, T; Choi, J; Kim, Y-H, Some identities on the [inlineequation not available: see fulltext.]-Bernstein polynomials, [inlineequation not available: see fulltext.]-Stirling numbers and [inlineequation not available: see fulltext.]-Bernoulli numbers, Advanced Studies in Contemporary Mathematics, 20, 335-341, (2010) · Zbl 1262.11020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.