Abstract
The main objective of this paper is to construct a new sequence of operators involving Apostol-Genocchi polynomials based on certain parameters. We investigate the rate of convergence of the operators given in this paper using second-order modulus of continuity and Voronovskaja type approximation theorem. Moreover, we find weighted approximation result of the given operators. Finally, we derive the Kantorovich variant of the given operators and discussed the approximation results.
Acknowledgement
The authors would like to thank the referee and editors for their many valuable comments and suggestions to improve the overall presentation of paper.
-
(Communicated by Gregor Dolinar)
References
[1] Abramowitz, M.—Stegun, I. A.: Handbook of Mathematical Functions, National Bureau of Standards, Washington, D. C., 1964.Search in Google Scholar
[2] Araci, S.: Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus, Appl. Math. Comput. 233 (2014), 599–607.10.1016/j.amc.2014.01.013Search in Google Scholar
[3] Claesson, A.—Kitaev, S.—Ragnarsson, K.—Tenner, B. E.: Boolean complexes for Ferrers graphs, Australas. J. Combin. 48 (2010), 159–173.Search in Google Scholar
[4] Chauhan, R.—Baxhaku, B.—Agrawal, P. N.: Szász Type Operators involving Charlier Polynomials of blending type, Complex Anal. Oper. Theory 13 (2019), 1197–1226.10.1007/s11785-018-0854-xSearch in Google Scholar
[5] De Vore, R. A.—Lorentz, G. G.: Constructive Approximation, Springer, Berlin 1993.10.1007/978-3-662-02888-9Search in Google Scholar
[6] Deo, N.—Dhamija, M.—Miclǎuş, D.: Stancu-Kantorovich operators based on inverse Pólya-Eggenberger distribution, Appl. Math. Comput. 273(1) (2016) 281-289.10.1016/j.amc.2015.10.008Search in Google Scholar
[7] Deo, N.—Dhamija, M.: Generalized positive linear operators based on PED and IPED, Iranian J. Sci. Technol. Trans. Sci. 43 (2019), 507–513.10.1007/s40995-017-0477-5Search in Google Scholar
[8] Gadzjiv, A. D.: Theorems of the type of P.P. Korovkin type theorems, Math. Zametki 20(5) (1976), 781–786; Math. Notes 20(5–6) (1976), 996–998 (English translation).Search in Google Scholar
[9] Gupta, V.—LÓpez-Moreno, A.-J.: Phillips operators preserving arbitrary exponential functions, eat, ebt, Filomat 32 (2018), 5071–5082.10.2298/FIL1814071GSearch in Google Scholar
[10] Gupta, V.—Agarwal, R. P.: Convergence Estimates in Approximation Theory, Springer International Publishing, Switzerland, 2014.10.1007/978-3-319-02765-4Search in Google Scholar
[11] Gupta, V.—Rassias, M. Th.: Moments of Linear Positive Operators and Approximation, Springer International Publishing, Switzerland, 2019.10.1007/978-3-030-19455-0Search in Google Scholar
[12] Ispir, N.—Aral, A.—Dogru, O.: On Kantorovich process of a sequence of generalized positive linear operators, Nonlinear Funct. Anal. Optimiz. 29(5–6) 2008, 574–589.10.1080/01630560802099365Search in Google Scholar
[13] Jolany, H.—Sharifi, H.—Alikelaye, R. E.: Some results for the Apostol-Genocchi polynomials of higher order, Bull. Malays. Math. Sci. Soc. 36(2) (2013), 465–479.Search in Google Scholar
[14] Jung, H. S.—Deo, N.—Dhamija, M.: Pointwise approximation by Bernstein type operators in mobile interval, Appl. Math. Comput. 214(1) (2014), 683–694.10.1016/j.amc.2014.07.034Search in Google Scholar
[15] Kajla, A.—Agrawal, P. N.: Szász-Kantorovich type operators based on Charlier polynomials, Kyungpook Math. J. 56(3) (2016), 877–897.10.5666/KMJ.2016.56.3.877Search in Google Scholar
[16] Kajla, A.—Agrawal, P. N.: Approximation properties of Szász type operators based on Charlier polynomials, Turkish J. Math. 39(6) (2015), 990–1003.10.3906/mat-1502-80Search in Google Scholar
[17] Korovkin, P. P.: On convergence of linear positive operators in the space of continuous functions (Russian), Doklady Akad. Nauk. SSSR (NS) 90 (1953), 961–964.Search in Google Scholar
[18] Luo, Q. M.: q-Extensions for the Apostol-Genocchi polynomials, Gen. Math. 17(2) (2009), 113–125.Search in Google Scholar
[19] Luo, Q. M.: Extensions of the Genocchi polynomials and their fourier expansions and integral representations, Osaka J. Math. 48 (2011), 291–309.Search in Google Scholar
[20] Luo, Q. M.—Srivastava, H. M.: Some generalizations of the-Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011), 5702–5728.10.1016/j.amc.2010.12.048Search in Google Scholar
[21] Natanson, I. P.: Constructive Function Theory, Vol. 1, Fredric Unger, New York, 1964.Search in Google Scholar
[22] Ozden, H.—Simsek, Y.: Modification and unification of the Apostol-type numbers and polynomials and their applications, Appl. Math. Comput. 235 (2014), 338–351.10.1016/j.amc.2014.03.004Search in Google Scholar
[23] Remmel, J.: Ascent Sequences, 2 + 2-free Posets, Upper Triangular Matrices, and Genocchi numbers, Workshop on Combinatorics, Enumeration, and Invariant Theory, George Mason University, Virginia, 2010.Search in Google Scholar
[24] Srivastava, H. M.—Özarslan, M. A.—Kaanoğlu, C.: Some generalized Lagrange-based Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Russ. J. Math. Phys. 20(1) (2013), 110–120.10.1134/S106192081301010XSearch in Google Scholar
[25] Varma, S.—Taşdelen, F.: Szász type operators involving Charlier polynomials, Math. Comput. Modell. 56 (2012), 118–122.10.1016/j.mcm.2011.12.017Search in Google Scholar
[26] Verma, D. K.—Gupta, V.: Approximation by a New Sequence of Operators Involving Charlier Polynomials with a Certain Parameter. In: Modern Mathematical Methods and High Performance Computing in Science and Technology, Springer, Singapore, 2016.10.1007/978-981-10-1454-3_3Search in Google Scholar
[27] Verma, D. K.—Gupta, V.—Agrawal, P. N.: Some approximation properties of Baskakov–Durrmeyer–Stancu operators, Appl. Math. Comput. 218(11) (2012), 6549–6556.10.1016/j.amc.2011.12.031Search in Google Scholar
[28] Weierstrass, K. G.: Über die analytische Darstellbarkeit sogenannter willkürlicher Funktionen einer reellen Veränderlichen, Sitzungsber. Akad. Berlin (1885), 633–639, 789–805.Search in Google Scholar
© 2021 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular papers
- Algebraic structures for pairwise comparison matrices: Consistency, social choices and Arrow’s theorem
- Polynomial functions on rings of dual numbers over residue class rings of the integers
- Sufficient conditions for p-valent functions
- Upper bounds for analytic summand functions and related inequalities
- Global structure for a fourth-order boundary value problem with sign-changing weight
- On the nonexistence conditions of solution of two-point in time problem for nonhomogeneous PDE
- Solvability of a nonlinear three-dimensional system of difference equations with constant coefficients
- Properties of critical and subcritical second order self-adjoint linear equations
- Korovkin type approximation via statistical e-convergence on two dimensional weighted spaces
- Approximation by a new sequence of operators involving Apostol-Genocchi polynomials
- Poisson like matrix operator and its application in p-summable space
- On the homological and algebraical properties of some Feichtinger algebras
- Disjoint topological transitivity for weighted translations generated by group actions
- On simultaneous limits for aggregation of stationary randomized INAR(1) processes with poisson innovations
- Marshall-Olkin Lindley-Log-logistic distribution: Model, properties and applications
- The shifted Gompertz-G family of distributions: Properties and applications
- On the testing hypothesis in uniform family of distributions with nuisance parameter
- Clarkson inequalities related to convex and concave functions
Articles in the same Issue
- Regular papers
- Algebraic structures for pairwise comparison matrices: Consistency, social choices and Arrow’s theorem
- Polynomial functions on rings of dual numbers over residue class rings of the integers
- Sufficient conditions for p-valent functions
- Upper bounds for analytic summand functions and related inequalities
- Global structure for a fourth-order boundary value problem with sign-changing weight
- On the nonexistence conditions of solution of two-point in time problem for nonhomogeneous PDE
- Solvability of a nonlinear three-dimensional system of difference equations with constant coefficients
- Properties of critical and subcritical second order self-adjoint linear equations
- Korovkin type approximation via statistical e-convergence on two dimensional weighted spaces
- Approximation by a new sequence of operators involving Apostol-Genocchi polynomials
- Poisson like matrix operator and its application in p-summable space
- On the homological and algebraical properties of some Feichtinger algebras
- Disjoint topological transitivity for weighted translations generated by group actions
- On simultaneous limits for aggregation of stationary randomized INAR(1) processes with poisson innovations
- Marshall-Olkin Lindley-Log-logistic distribution: Model, properties and applications
- The shifted Gompertz-G family of distributions: Properties and applications
- On the testing hypothesis in uniform family of distributions with nuisance parameter
- Clarkson inequalities related to convex and concave functions