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A study on the \(p\)-adic integral representation on \(\mathbb Z_p\) associated with Bernstein and Bernoulli polynomials. (English) Zbl 1221.11058

The authors study Bernstein polynomials on \(\mathbb Z_{p}\) and investigate certain properties of Bernstein polynomials related to Stirling numbers and Bernoulli numbers.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11B73 Bell and Stirling numbers
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
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[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 · Zbl 1002.11088
[2] Acikgoz, M.; Araci, S., On the generating function of the Bernstein polynomials (2010), Rhodes, Greece
[3] Bernstein S: Demonstration du theoreme de Weierstrass, fondee sur le calcul des probabilities.Communications of the Kharkov Mathematical Society 1913, 13: 1-2.
[4] Kim, T.; Jang, LC; Yi, H., A note on the modified q-Bernstein polynomials, No. 2010, 12 (2010) · Zbl 1198.33005
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[10] Kim T: On aq-analogue of thep-adic log gamma functions and related integrals.Journal of Number Theory 1999,76(2):320-329. 10.1006/jnth.1999.2373 · Zbl 0941.11048 · doi:10.1006/jnth.1999.2373
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[12] Kim T: Power series and asymptotic series associated with theq-analog of the two-variablep-adicL-function.Russian Journal of Mathematical Physics 2005,12(2):186-196. · Zbl 1190.11049
[13] Kim T, Choi J, Kim Y-H: Some identities on the q-Bernstein polynomials, g-Stirling numbers and q-Bernoulli numbers.Advanced Studies in Contemporary Mathematics 2010,20(3):335-341. · Zbl 1262.11020
[14] Kim T: Some identities on theq-Euler polynomials of higher order andq-Stirling numbers by the fermionicp-adic integral on[InlineEquation not available: see fulltext.].Russian Journal of Mathematical Physics 2009,16(4):484-491. 10.1134/S1061920809040037 · Zbl 1192.05011 · doi:10.1134/S1061920809040037
[15] Kim T: Onp-adic interpolating function forq-Euler numbers and its derivatives.Journal of Mathematical Analysis and Applications 2008,339(1):598-608. 10.1016/j.jmaa.2007.07.027 · Zbl 1160.11013 · doi:10.1016/j.jmaa.2007.07.027
[16] Kim T, Park D-W, Rim S-H: On multivariatep-adicq-integrals.Journal of Physics A 2001,34(37):7633-7638. 10.1088/0305-4470/34/37/315 · Zbl 1002.11088 · doi:10.1088/0305-4470/34/37/315
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