A Note on Uniformly Convergence for Positive Linear Operators Involving Euler Type Polynomials

Abstract

Positive linear operators play a significant role in many domains, particularly numerical and mathematical analysis. Specifically, they are commonly found in a wide variety of methods to resolve optimization and differential equation issues. Basic properties of positive linear operators are linearity, positivity, positive linear and being restrictive. There are various ways to examine the significance of positive linear operators in Approximation Theory. The Convergence Analysis is the most significant of these. In many situations involving numerical analysis and convergence analysis, positive linear operators are essential. Positive linear operators must be able to converge in iterations towards a specific goal, especially in various approximation techniques or iterative solution algorithms. This can be used to solve optimization issues more effectively or to increase the precision of numerical answers. In approximation theory, generating functions are essential. They are specifically used to build algorithms that facilitate the proper approximation to a goal and to examine the approximation in question. The speed at which an approximation converges to a target can also be ascertained via generating functions. An essential tool for evaluating and enhancing the rate of convergence of iterative algorithms is offered by these functions. The aim of this study is to construct a generalized Kantorovich type Szász operators including the generating functions of Euler polynomials with order (-1). Moreover, we derive the moment and central moment functions for these operators. Finally, we show uniformly convergence of operators by using Korovkin theorem.

Keywords

• Positive linear operators
• Generating functions
• Euler type polynomials
• Korovkin Theorem
• Moment functions

References

1. Atakut, Ç., and Büyükyazıcı, İ. (2016). Approximation by Kantorovich-Szász type operators based on Brenke type polynomials. Numerical Functional Analysis and Optimization, 37(12), 1488-1502.
2. Gezer, K., and Yılmaz, M. M. (2023). Approximation properties of a class of Kantorovich type operators associated with the Charlier polynomials. Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 28(2), 383-393.
3. Gupta, V., and Rassias, M. T. (2019). Moments of linear positive operators and approximation. Switzerland: Springer International Publishing.
4. Horadam, A. F. (1992). Negative order Genocchi polynomials. Fibonacci Q, 30, 21-34.
5. İçöz, G., Varma, S., and Sucu, S. (2016). Approximation by operators including generalized Appell polynomials. Filomat, 30(2), 429-440.
6. Kilar, N., and Simsek, Y. (2021). Formulas and relations of special numbers and polynomials arising from functional equations of generating functions. Montes Taurus Journal of Pure and Applied Mathematics, 3(1), 106-123.
7. McBride, E. B. (2012). Obtaining generating functions (Vol. 21). Springer Science and Business Media.
8. Özarslan, M. A., Duman, O., and Srivastava, H. M. (2008). Statistical approximation results for Kantorovich-type operators involving some special polynomials. Mathematical and Computer Modelling, 48(3-4), 388-401.

9. Paltanea, R. (2012). Approximation theory using positive linear operators. Springer Science and Business Media.
10. Simsek, Y. (2018). New families of special numbers for computing negative order Euler numbers and related numbers and polynomials. Applicable Analysis and Discrete Mathematics, 12(1), 1-35.
11. Sofyalıoğlu, M., and Kanat, K. (2022). Approximation by Szász-Baskakov operators based on Boas-Buck-type polynomials. Filomat, 36(11), 3655-3673.
12. Srivastava, H. M., Cao, J., and Arjika, S. (2020). A note on generalized q-difference equations and their applications involving q-hypergeometric functions. Symmetry, 12(11), 1816.
13. Taşdelen, F., Aktaş, R., and Altın, A. (2012, January). A Kantorovich type of Szász operators including Brenke-type polynomials. Abstract and Applied Analysis, 2012. https://doi.org/10.1155/2012/867203
14. Yılmaz, M. M. (2022). Approximation by Szász type operators involving Apostol–Genocchi polynomials. Computer Modeling in Engineering and Sciences, 130(1), 287-297.
15. Yılmaz, M. M. (2023). Rate of convergence by Kantorovich type operators involving adjoint Bernoulli polynomials. Publications de l'Institut Mathematique, 114(128), 51-62.