Approximation Properties of a Fractional Integral-Type Szász–Kantorovich–Stancu–Schurer Operator via Charlier Polynomials

Rao, Nadeem; Farid, Mohammad; Jha, Nand Kishor

doi:10.3390/math13183039

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Approximation Properties of a Fractional Integral-Type Szász–Kantorovich–Stancu–Schurer Operator via Charlier Polynomials

by

Nadeem Rao

¹

,

Mohammad Farid

^2,*

and

Nand Kishor Jha

³

¹

Department of Mathematics, University Center for Research and Development, Chandigarh University, Mohali 140413, Punjab, India

²

Department of Mathematics, College of Science, Qassim University, Qassim 52571, Saudi Arabia

³

Department of Mathematics, Chandigarh University, Mohali 140413, Punjab, India

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(18), 3039; https://doi.org/10.3390/math13183039

Submission received: 25 July 2025 / Revised: 9 September 2025 / Accepted: 17 September 2025 / Published: 20 September 2025

(This article belongs to the Special Issue Advances in Functional Analysis and Approximation Theory)

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Abstract

The goal of this manuscript is to introduce a new Stancu generalization of the modified Szász–Kantorovich operator connecting Riemann–Liouville fractional operators via Charlier polynomials. Further, some estimates are calculated as test functions and central moments. In the next section, we investigate some convergence analysis along with the rate of approximations. Moreover, we discuss the order of approximation of a higher-order modulus of smoothness with the help of some moments and establish some convergence results concerning Peetre’s K-functional, Lipschitz-type functions for a newly developed operator

S K_{n + p, a}^{v_{1}, v_{2}}

. We estimate some results related to Korovkin-, Voronovskaya-, and Grüss–Voronovskaya-type theorems.

Keywords: rate of convergence; order of approximation; mathematical operators; modulus of continuity; approximation algorithms; Peetre`s K-functional; Korovkin theorem

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MDPI and ACS Style

Rao, N.; Farid, M.; Jha, N.K. Approximation Properties of a Fractional Integral-Type Szász–Kantorovich–Stancu–Schurer Operator via Charlier Polynomials. Mathematics 2025, 13, 3039. https://doi.org/10.3390/math13183039

AMA Style

Rao N, Farid M, Jha NK. Approximation Properties of a Fractional Integral-Type Szász–Kantorovich–Stancu–Schurer Operator via Charlier Polynomials. Mathematics. 2025; 13(18):3039. https://doi.org/10.3390/math13183039

Chicago/Turabian Style

Rao, Nadeem, Mohammad Farid, and Nand Kishor Jha. 2025. "Approximation Properties of a Fractional Integral-Type Szász–Kantorovich–Stancu–Schurer Operator via Charlier Polynomials" Mathematics 13, no. 18: 3039. https://doi.org/10.3390/math13183039

APA Style

Rao, N., Farid, M., & Jha, N. K. (2025). Approximation Properties of a Fractional Integral-Type Szász–Kantorovich–Stancu–Schurer Operator via Charlier Polynomials. Mathematics, 13(18), 3039. https://doi.org/10.3390/math13183039

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Approximation Properties of a Fractional Integral-Type Szász–Kantorovich–Stancu–Schurer Operator via Charlier Polynomials

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