# Generalized Statistical Convergence Based on Fractional Order Difference Operator and Its Applications to Approximation Theorems

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## Abstract

In this paper, following very recent results of Baliarsingh (Alex Eng J 55(2):1811–1816, 2016), we first introduce the concepts of statistically \(\Omega ^{\Delta }\)-summability and \(\Omega ^{\Delta }\)-statistical convergence by means of fractional-order difference operator \(\Delta ^{\alpha ,\beta ,\gamma }_{h}\). We also present some important inclusion relations between newly proposed methods. Our present investigation deals essentially with various summability techniques and reveals how these methods lead to a number of approximation by positive linear operators. As an application, we prove a Korovkin type approximation theorem and also present an illustrative example using the generating function type Meyer-König and Zeller operator. Furthermore, we estimate the rate of convergence of approximating linear operators by means of the modulus of continuity and some Voronovskaja type results are derived. Finally, we present some computational and geometrical interpretations to illustrate some of our approximation results.

## Keywords

Statistical convergence and statistical summability Fractional order difference operator Korovkin and Voronovskaja type approximation theorems Modulus of continuity and rate of convergence Meyer-König and Zeller polynomials## Mathematics Subject Classification

Primary 40C05 40G15 41A36 Secondary 46A35 46A45 46B45## References

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