Convergence Properties of a Kantorovich Type of Szász Operators Involving Negative Order Genocchi Polynomials

Year 2023, Volume: 10 Issue: 2, 196 - 205, 27.06.2023

Erkan Ağyüz

https://doi.org/10.54287/gujsa.1282992

Abstract

The goal of this research is to construct a generalization of a Kantorovich type of Szász operators involving negative-order Genocchi polynomials. With the aid of Korovkin’s theorem, modulus of continuity, Lipschitz class, and Peetre’s K-functional the approximation properties and convergence rate of these operators are established. To illustrate how operators converge to a certain function, we present some examples.

Keywords

Generating Functions , KorovkinType Approximation , Modulus of Continuity , Genocchi Polynomials

References

• Agyuz, E. (2021a). On The Convergence Properties of Kantorovich-Szász Type Operators Involving Tangent Polynomials. Adıyaman University Journal of Science, 11(2), 244-252. doi:10.37094/adyujsci.905311
• Agyuz, E. (2021b, November 11-12). Convergence by Szász type operators based on Euler type polynomials. In: The 3rd & 4th Mediterranean International Conference of Pure & Applied Mathematics and Related Areas (MICOPAM 2020-2021). Antalya, Türkiye.
• Agyuz, E. (2022). A Study on Kantorovich Type Operator Involving Adjoint Euler Polynomials. Conference Proceedings of Science and Technology, 5(1),178-181.
• Agyuz, E. (2023). A Generalization of Szász Type Operators Involving Generating Function of Negative Order Genocchi Polynomials. In: A. Akpınar (Eds.), Research on Mathematics and Science, (pp. 15-26). doi:10.58830/ozgur.pub81
• Altomare, F. (2010). Korovkin-type theorems and approximation by positive linear operators. Survey Approx. Theory, 5, 92-164.
• Atakut, Ç., & Büyükyazıcı, İ. (2016). Approximation by Kantorovich-Szász type operators based on Brenke type polynomials. Numerical Functional Analysis and Optimization, 37(12), 1488-1502. doi:10.1080/01630563.2016.1216447
• Cangul, İ. N., Ozden, H., & Simsek, Y. (2009) A new approach to q-Genocchi numbers and their interpolation functions. Nonlinear Analysis: Theory, Methods & Applications, 71(12), e793-e799. doi:10.1016/j.na.2008.11.040
• Davis, P. J. (1975). Interpolation and approximation. Courier Corporation.
• DeVore, R. A., & Lorentz, G. G. (1993). Constructive approximation (Vol. 303). Springer, Berlin.
• Gupta, V., & Rassias, M. T. (2019). Moments of linear positive operators and approximation. Switzerland: Springer International Publishing. doi:10.1007/978-3-030-19455-0
• Horadam, A. F. (1992). Negative order Genocchi polynomials. Fibonacci Q, 30, 21-34.
• İçöz, G., Varma, S., & Sucu, S. (2016). Approximation by operators including generalized Appell polynomials. Filomat, 30(2), 429-440. doi:10.2298/FIL1602429I
• Jakimovski, A., & Leviatan, D. (1969). Generalized Szász operators for the approximation in the infinite interval. Mathematica (Cluj), 11(34), 97-103.
• Kilar, N., & Simsek, Y. (2020). Formulas involving sums of powers, special numbers and polynomials arising from p-adic integrals, trigonometric and generating functions. Publications de l'Institut Mathematique, 108(122), 103-120. doi:10.2298/PIM2022103K
• Kilar, N., & 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.
• Korovkin, P. P. (1953). Convergence of linear positive operators in the spaces of continuous functions (Russian). Doklady Akad. Nauk. SSSR (N.S.), 90, 961-964.
• Korovkin, P. P. (1960). Linear Operators and Approximation Theory. Translated from the Russian ed. (1959), Russian Monographs and Texts on Advances Mathematics and Physics, Vol. III. Gordon and Breach Publishers, Inc. New York, Hindustan Publ. Corp. (India), Delhi.
• Kucukoglu, I., Simsek, B., & Simsek, Y. (2019). An approach to negative hypergeometric distribution by generating function for special numbers and polynomials. Turkish Journal of Mathematics, 43(5), 2337-2353. doi:10.3906/mat-1906-6
• Kucukoglu, I. (2022). Computational and implementational analysis of generating functions for higher order combinatorial numbers and polynomials attached to Dirichlet characters. Mathematical Methods in the Applied Sciences, 45(9), 5043-5066. doi:10.1002/mma.8092
• Lupas, A. (1995, March 13-17). The approximation by some positive linear operators. In: M. W. Müller, M. Felten, D. H. Mache. (Eds.), Proceedings of the International Dortmund Meeting on Approximation Theory (pp. 201-229). Witten, Germany.
• Menekşe Yılmaz, M. (2022). Approximation by Szasz Type Operators Involving Apostol-Genocchi Polynomials. CMES-Computer Modeling in Engineering & Sciences, 130(1), 287-297. doi:10.32604/cmes.2022.017385
• Mursaleen, M., Al-Abied, A. A. H., & ACU, A. M. (2018). Approximation by Chlodowsky type of Szasz operators based on Boas--Buck-type polynomials. Turkish Journal of Mathematics, 42(5), 2243-2259. doi:10.3906/mat-1803-62
• Simsek, Y. (2008). Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions. Advanced Studies in Contemporary Mathematics, 16(2), 251-278.
• Simsek, Y. (2012). Generating functions for q-Apostol type Frobenius–Euler numbers and polynomials. Axioms, 1(3), 395-403. doi:10.3390/axioms1030395
• Simsek, Y. (2013). Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications. Fixed Point Theory and Applications, 2013(1), 87. doi:10.1186/1687-1812-2013-87
• Simsek, Y. (2017). Computation methods for combinatorial sums and Euler-type numbers related to new families of numbers. Mathematical Methods in the Applied Sciences, 40(7), 2347-2361. doi:10.1002/mma.4143
• 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. doi:10.2298/AADM1801001S
• Srivastava, H. M., & Choi, J. (2001). Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht, Boston and London.
• Srivastava, H. M., Kurt, B., & Simsek, Y. (2012). Some families of Genocchi type polynomials and their interpolation functions. Integral Transforms and Special Functions, 23(12), 919-938. doi:10.1080/10652469.2011.643627
• Varma, S., Sucu, S., & İçöz, G. (2012). Generalization of Szász operators involving Brenke type polynomials. Computers & Mathematics with Applications, 64(2), 121-127. doi:10.1016/j.camwa.2012.01.025