A new generating function of (\(q\)-) Bernstein-type polynomials and their interpolation function. (English) Zbl 1185.33013

Summary: The main object of this paper is to construct a new generating function of the (\(q\)-) Bernstein-type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relations and derivatives of the (\(q\)-) Bernstein-type polynomials. We also give relations between the (\(q\)-) Bernstein-type polynomials, Hermite polynomials, Bernoulli polynomials of higher order, and Stirling numbers of the second kind. By applying Mellin transformation to this generating function, we define interpolation of the (\(q\)-) Bernstein-type polynomials. Moreover, we give some applications and questions on approximations of (\(q\)-) Bernstein-type polynomials, and moments of some distributions in Statistics.


33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
05A15 Exact enumeration problems, generating functions
11B73 Bell and Stirling numbers
11B68 Bernoulli and Euler numbers and polynomials
41A10 Approximation by polynomials
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[1] S. N. Bernstein, “Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités,” Communications of the Mathematical Society of Charkow. Séries 2, vol. 13, pp. 1-2, 1912-1913.
[2] M. Acikgoz and Y. Simsek, “On multiple interpolation functions of the Nörlund-type q-Euler polynomials,” Abstract and Applied Analysis, vol. 2009, Article ID 382574, 14 pages, 2009. · Zbl 1194.11027 · doi:10.1155/2009/382574
[3] N. P. Cakić and G. V. Milovanović, “On generalized Stirling numbers and polynomials,” Mathematica Balkanica, vol. 18, no. 3-4, pp. 241-248, 2004. · Zbl 1177.05014
[4] S. Berg, “Some properties and applications of a ratio of Stirling numbers of the second kind,” Scandinavian Journal of Statistics, vol. 2, no. 2, pp. 91-94, 1975. · Zbl 0312.62080
[5] T. N. T. Goodman, H. Oru\cc, and G. M. Phillips, “Convexity and generalized Bernstein polynomials,” Proceedings of the Edinburgh Mathematical Society, vol. 42, no. 1, pp. 179-190, 1999. · Zbl 0930.41010 · doi:10.1017/S0013091500020101
[6] H. W. Gould, “A theorem concerning the Bernstein polynomials,” Mathematics Magazine, vol. 31, no. 5, pp. 259-264, 1958. · Zbl 0088.27301 · doi:10.2307/3029385
[7] Z. Guan, “Iterated Bernstein polynomial approximations,” http://arxiv.org/abs/0909.0684.
[8] A. H. Joarder and M. Mahmood, “An inductive derivation of Stirling numbers of the second kind and their applications in statistics,” Journal of Applied Mathematics & Decision Sciences, vol. 1, no. 2, pp. 151-157, 1997. · Zbl 0910.62015 · doi:10.1155/S1173912697000138
[9] K. I. Joy, “Bernstein polynomials, On-Line Geometric Modeling Notes,” http://en.wikipedia.org/wiki/Bernstein_polynomial.
[10] E. Kowalski, “Bernstein polynomials and Brownian motion,” American Mathematical Monthly, vol. 113, no. 10, pp. 865-886, 2006. · Zbl 1149.60051 · doi:10.2307/27642086
[11] N. E. Nörlund, Vorlesungen über Differenzenrechung, Springer, Berlin, Germany, 1924.
[12] J. L. López and N. M. Temme, “Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials,” Journal of Mathematical Analysis and Applications, vol. 239, no. 2, pp. 457-477, 1999. · Zbl 0979.33004 · doi:10.1006/jmaa.1999.6584
[13] G. Nowak, “Approximation properties for generalized q-Bernstein polynomials,” Journal of Mathematical Analysis and Applications, vol. 350, no. 1, pp. 50-55, 2009. · Zbl 1162.33009 · doi:10.1016/j.jmaa.2008.09.003
[14] H. Oru\cc and G. M. Phillips, “A generalization of the Bernstein polynomials,” Proceedings of the Edinburgh Mathematical Society, vol. 42, no. 2, pp. 403-413, 1999. · Zbl 0930.41009 · doi:10.1017/S0013091500020332
[15] H. Oru\cc and N. Tuncer, “On the convergence and iterates of q-Bernstein polynomials,” Journal of Approximation Theory, vol. 117, no. 2, pp. 301-313, 2002. · Zbl 1015.33012 · doi:10.1006/jath.2002.3703
[16] S. Ostrovska, “The approximation by q-Bernstein polynomials in the case q\downarrow 1,” Archiv der Mathematik, vol. 86, no. 3, pp. 282-288, 2006. · Zbl 1096.41004 · doi:10.1007/s00013-005-1503-y
[17] S. Ostrovska, “On the q-Bernstein polynomials,” Advanced Studies in Contemporary Mathematics, vol. 11, no. 2, pp. 193-204, 2005. · Zbl 1116.41013
[18] G. M. Phillips, “Bernstein polynomials based on the q-integers, the heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin,” Annals of Numerical Mathematics, vol. 4, no. 1-4, pp. 511-518, 1997. · Zbl 0881.41008
[19] G. M. Phillips, Interpolation and Approximation by Polynomials, vol. 14 of CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, NY, USA, 2003. · Zbl 1023.41002 · doi:10.1007/b97417
[20] G. M. Phillips, “A survey of results on the q-Bernstein polynomials,” IMA Journal of Numerical Analysis, vol. 30, no. 1, pp. 277-288, 2010. · Zbl 1191.41002 · doi:10.1093/imanum/drn088
[21] Á. Pintér, “On a Diophantine problem concerning Stirling numbers,” Acta Mathematica Hungarica, vol. 65, no. 4, pp. 361-364, 1994. · Zbl 0811.11017 · doi:10.1007/BF01876037
[22] Y. Simsek, “Twisted (h,q)-Bernoulli numbers and polynomials related to twisted (h,q)-zeta function and L-function,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 790-804, 2006. · Zbl 1139.11051 · doi:10.1016/j.jmaa.2005.12.057
[23] Y. Simsek, “On q-deformed Stirling numbers,” http://arxiv.org/abs/0711.0481.
[24] Y. Simsek, V. Kurt, and D. Kim, “New approach to the complete sum of products of the twisted (h,q)-Bernoulli numbers and polynomials,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 44-56, 2007. · Zbl 1163.11015 · doi:10.2991/jnmp.2007.14.1.5
[25] Z. Wu, “The saturation of convergence on the interval [0,1] for the q-Bernstein polynomials in the case q>1,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 137-141, 2009. · Zbl 1236.41011 · doi:10.1016/j.jmaa.2009.04.003
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