Generalized Statistically Almost Convergence Based on the Difference Operator which Includes the (p, q)-Gamma Function and Related Approximation Theorems

Abstract

This paper is devoted to extend the notion of almost convergence and its statistical forms with respect to the difference operator involving (p, q)-gamma function and an increasing sequence $$(\lambda _n)$$ of positive numbers. We firstly introduce some new concepts of almost $${\Delta }^{[a,b,c]}_{h, \alpha , \beta }(\lambda )$$ -statistical convergence, statistical almost $${\Delta }^{[a,b,c]}_{h, \alpha , \beta }(\lambda )$$ -convergence and strong almost $$[{\Delta }^{[a,b,c]}_{h, \alpha , \beta }(\lambda )]_r$$ -convergence. Moreover, we present some inclusion relations between these newly proposed methods and give some counterexamples to show that these are non-trivial generalizations of existing literature on this topic. We then prove a Korovkin type approximation theorem for functions of two variables through statistically almost $${\Delta }^{[a,b,c]}_{h, \alpha , \beta }(\lambda )$$ -convergence and also present an illustrative example via bivariate non-tensor type Meyer–Konig and Zeller generalization of Bernstein power series. Furthermore, we estimate the rate of almost convergence of approximating linear operators by means of the modulus of continuity and derive some Voronovskaja type results by using the generalized Meyer–Konig and Zeller operators. Finally, some computational and geometrical interpretations for the convergence of operators to a function are presented.