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On the fermionic \(p\)-adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials. (English) Zbl 1239.11020

The authors prove formulas with Euler numbers, as application of properties of Bernstein polynomials.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
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[1] Kim T: [InlineEquation not available: see fulltext.]-Volkenborn integration. Russian Journal of Mathematical Physics 2002, 9(3):288-299. · Zbl 1092.11045
[2] Kim, T., Barnes-type multiple [InlineEquation not available: see fulltext.]-zeta functions and [InlineEquation not available: see fulltext.]-Euler polynomials (2010)
[3] 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
[4] Bernstein S: Démonstration du théorème de Weierstrass, fondée sur le calcul des probabilities. Communications of the Kharkov Mathematical Society 1912, 13: 1-2.
[5] Kim T, Choi J, Kim YH: On extended carlitz’s type [InlineEquation not available: see fulltext.]-Euler numbers and polynomials. Advanced Studies in Contemporary Mathematics 2010, 20(4):499-505. · Zbl 1244.05029
[6] Govil NK, Gupta V: Convergence of [InlineEquation not available: see fulltext.]-Meyer-König-Zeller-Durrmeyer operators. Advanced Studies in Contemporary Mathematics 2009, 19(1):97-108. · Zbl 1182.41011
[7] Gupta V, Kim T, Choi J, Kim Y-H: Generating function for [InlineEquation not available: see fulltext.]-Bernstein, [InlineEquation not available: see fulltext.]-Meyer-König-Zeller and [InlineEquation not available: see fulltext.]-beta basis. Automation Computers Applied Mathematics 2010, 19: 7-11.
[8] Kim T: [InlineEquation not available: see fulltext.]-extension of the Euler formula and trigonometric functions. Russian Journal of Mathematical Physics 2007, 14(3):275-278. 10.1134/S1061920807030041 · Zbl 1188.33001 · doi:10.1134/S1061920807030041
[9] Kim T: [InlineEquation not available: see fulltext.]-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 2008, 15(1):51-57. · Zbl 1196.11040 · doi:10.1134/S1061920808010068
[10] 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 2010, 20(3):335-341. · Zbl 1262.11020
[11] Kim, T.; Jang, L-C; Yi, H., A note on the modified [InlineEquation not available: see fulltext.]-bernstein polynomials, No. 2010 (2010)
[12] Kim T: Note on the Euler [InlineEquation not available: see fulltext.]-zeta functions. Journal of Number Theory 2009, 129(7):1798-1804. 10.1016/j.jnt.2008.10.007 · Zbl 1221.11231 · doi:10.1016/j.jnt.2008.10.007
[13] Kurt V: A further symmetric relation on the analogue of the Apostol-Bernoulli and the analogue of the Apostol-Genocchi polynomials. Applied Mathematical Sciences 2009, 3(53-56):2757-2764. · Zbl 1269.11024
[14] Cangul IN, Kurt V, Ozden H, Simsek Y: On the higher-order [InlineEquation not available: see fulltext.]-[InlineEquation not available: see fulltext.]-Genocchi numbers. Advanced Studies in Contemporary Mathematics 2009, 19(1):39-57. · Zbl 1187.05004
[15] Jang, L-C; Kim, W-J; Simsek, Y., A study on the [InlineEquation not available: see fulltext.]-adic integral representation on [InlineEquation not available: see fulltext.] associated with Bernstein and Bernoulli polynomials, No. 2010 (2010)
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