Identities involving values of Bernstein, \(q\)-Bernoulli, and \(q\)-Euler polynomials. (English) Zbl 1256.11013

Bayad, A.; Kim, Taekyun

Russ. J. Math. Phys. 18, No. 2, 133-143 (2011).

Summary: In this paper, we give relations involving values of \(q\)-Bernoulli, \(q\)-Euler, and Bernstein polynomials. Using these relations, we obtain some interesting identities on the \(q\)-Bernoulli, \(q\)-Euler, and Bernstein polynomials.

MSC:

11B68	Bernoulli and Euler numbers and polynomials
33D45	Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
05A30	\(q\)-calculus and related topics

References:

[1]	M. Acikgoz and S. Araci, ”A Study on the Integral of the Product of Several Type Berstein Polynomials,” IST Trans. Appl. Math.-Model. Simulat., 11 (2), 10–14 (2010).
[2]	A. Bayad, T. Kim, B. Lee, and S.-H. Rim, ”Some Identities on the Bernstein Polynomials Associated with q-Euler Polynomials” (Communicated). · Zbl 1234.11024
[3]	S. Bernstein, ”Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités,” Commun. Soc. Math. Kharkow 13, 1–2 (1912).
[4]	I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, ”On the Higher-Order w-q-Genocchi Numbers,” Adv. Stud. Contemp. Math. 19, 39–57 (2009). · Zbl 1187.05004
[5]	H. W. Gould, ”A Theorem Concerning the Bernstein Polynomials,” Math. Mag., 31(5), 259–264 (1958). · Zbl 0088.27301
[6]	N. K. Govil and V. Gupta, ”Convergence of q-Meyer-König-Zeller-Durrmeyer Operators,” Adv. Stud. Contemp. Math. 19, 97–108 (2009). · Zbl 1182.41011
[7]	L.C. Jang, W-J. Kim, and Y. Simsek, ”A Study on the p-Adic Integral Representation on \(\mathbb{Z}\)p Associated with Bernstein and Bernoulli Polynomials,” Adv. Difference Equ., (2010), Article ID 163219, 6pp. · Zbl 1221.11058
[8]	K. I. Joy, ”Bernstein Polynomials,” On-Line Geometric Modelling Notes, http://en.wikipedia.org/wiki/Bernsteinpolynomials .
[9]	T. Kim, ”Barnes-Type Multiple q-Zeta Functions and q-Euler Polynomials,” J. Phys. A: Math. Theor. 43 255201, (2010), 11p. · Zbl 1213.11050
[10]	T. Kim, ”An Analogue of Bernoulli Numbers and Their Congruences,” Rep. Fac. Sci. Engrg. Saga Univ. Math. 22(2), 21–26 (1994). · Zbl 0802.11007
[11]	T. Kim, ”On the Symmetries of the q-Bernoulli Polynomials,” Abstr. Appl. Anal., (2008), Article ID 914367, 7pp. · Zbl 1217.11022
[12]	T. Kim, ”A New Approach to p-Adic q-L-Functions,” Adv. Stud. Contemp. Math. 12(1), 61–72 (2006). · Zbl 1084.11065
[13]	T. Kim, ”q-Bernoulli Numbers and Polynomials Associated with Gaussian Binomial Coefficients,” Russ. J. Math. Phys. 15, 51–59 (2008). · Zbl 1196.11040
[14]	T. Kim, J. Choi, and Y. H. Kim, ”q-Bernstein Polynomials Associated with q-Stirling Numbers and Carlitz’s q-Bernoulli Numbers,” Absr. Appl. Anal. (2010), Article ID 150975, 11. · Zbl 1208.11029
[15]	T. Kim, ”Some Identities on the q-Euler Polynomials of Higner Order and q-Stirling Numbers by Fermionic p-Adic Integral on \(\mathbb{Z}\)p,” Russ. J. Math. Phys. 16, 484–491 (2009). · Zbl 1192.05011
[16]	T. Kim, ”Note on the Euler q-Zeta Functions,” J. Number Theory 129, 1798–1804 (2009). · Zbl 1221.11231
[17]	T. Kim and B. Lee, ”Some Identities of the Frobenius-Euler Polynomials,” Abstr. Appl. Anal. (2009), Article ID 639439. · Zbl 1246.11052
[18]	T. Kim, J. Choi, Y.H. Kim, and C. S. Ryoo, ”On the Fermionic p-Adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials,” J. Inequal. and Appl. (2010), Article ID 864249, 12. · Zbl 1239.11020
[19]	B. A. Kupershmidt, ”Reflection Symmetries of q-Bernoulli Polynomials,” J. Nonlinear Math. Phys. 12, 412–422 (2005). · Zbl 1362.33021
[20]	G. G. Lorentz, Bernstein Polynomials (2nd edition) (printed in USA 1986).
[21]	S-H. Rim, J-H. Jin, E-J. Moon, and S-J. Lee, ”On Multiple Interpolation Functions of the q-Genocchi Polynomials,” J. Inequal. Appl., (2010), Article ID 351419, 13.
[22]	S. Zorlu, H. Aktuglu, and M. A. Ozarslan, ”An Estimation to the Solution of an Initial Value Problem via q-Bernstein Polynomials,” J. Comput. Anal. Appl. 12(3), 637–645 (2010). · Zbl 1197.41012