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A Study of Generating Functions Related to New Bernstein-Type Operators


Authors : Raksha Rani Agrawal

Volume/Issue : Volume 11 - 2026, Issue 5 - May


Google Scholar : https://tinyurl.com/puse3w6j

Scribd : https://tinyurl.com/2k85xcs7

DOI : https://doi.org/10.38124/ijisrt/26May2086

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Abstract : The objective of this paper is to develop a generating function for a novel class of Bernstein type operators, as proposed by N. Deo et al. ([3]). This involves deriving functional equations and partial differentiation’s based on the underlying basis functions. In addition, we investigate the use of the Laplace transform to generate these new Bernstein basis functions.

Keywords : Bernstein Polynomials; Generating Functions; Functional Equations.

References :

  1. Ac´ıkg¨oz, M., Araci, S. (2010, September). On the generating function for Bernstein polynomials. In AIP Conference Proceedings (Vol. 1281, No. 1, pp. 1141-1143). American Institute of Physics.
  2. Acikgoz, M., Simsek, Y. (2009, January). On multiple interpolation functions of the N¨orlund-type-Euler polynomials. In Abstract and Applied Analysis (Vol. 2009). Hindawi.
  3. Deo, N., Noor, M. A., Siddiqui, M. A. (2008). On approximation by a class of new Bernstein type operators. Applied mathematics and computation, 201(1-2), 604-612.
  4. Farouki, R., Goodman, T. (1996). On the optimal stability of the Bernstein basis. Mathematics of computation, 65(216), 1553-1566.
  5. Goldman, R. N. (1995). Identities for the univariate and bivariate Bernstein basis functions. In Graphics gems V (pp. 149-162). Academic Press.
  6. Gould, H. W. (1958). A theorem concerning the Bernstein polynomials. Mathematics Magazine, 31(5), 259-264.
  7. Guan, Z. (2009). Iterated Bernstein polynomial approximations. arXiv preprint arXiv:0909.0684.
  8. Kim, T., Rim, S. H., Simsek, Y., Kim, D. (2008). On the analogs of Bernoulli and Euler numbers, related identities and zeta and L-functions. J. Korean Math. Soc, 45(2), 435-453.
  9. Lorentz, G. G. (2012). Bernstein polynomials. American Mathematical Soc.
  10. Simsek, Y. (2013). Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions. Fixed Point Theory and Applications, 2013(1), 1-13.
  11. S˙IMSEK, Y. (2014). Generating functions for the Bernstein type polynomials: a new approach to deriving identities and applications for the polynomials. Hacettepe Journal of Mathematics and Statistics, 43(1), 1-14.
  12. Simsek, Y., Acikgoz, M. (2010, January). A New Generating Function of Bernstein-Type Polynomials and Their Interpolation Function. In Abstract and Applied Analysis (Vol. 2010). Hindawi.
  13. Simsek, Y. (2013). q-Beta polynomials and their applications. Applied Mathematics Information Sciences, 7(6).
  14. Simsek, Y. (2006). Twisted (h, q)-Bernoulli numbers and polynomials related to twisted (h, q)-zeta function and L-function. Journal of Mathematical Analysis and Applications, 324(2), 790-804.
  15. Srivastava, H. M., Kim, T., Simsek, Y. (2005). q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series. Russian Journal of Mathematical Physics, 12(2), 241-268.

The objective of this paper is to develop a generating function for a novel class of Bernstein type operators, as proposed by N. Deo et al. ([3]). This involves deriving functional equations and partial differentiation’s based on the underlying basis functions. In addition, we investigate the use of the Laplace transform to generate these new Bernstein basis functions.

Keywords : Bernstein Polynomials; Generating Functions; Functional Equations.

Paper Submission Last Date
30 - June - 2026

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