On Approximation Properties of Generalized Lupa\c{s} Type Operators Based on Polya Distribution with Pochhammer $k$-Symbol

TL;DR

This paper introduces a generalized Kantorovich-Lupa extcyr{ }-Stancu operator based on Polya distribution with Pochhammer $k$-symbol, analyzing its convergence, rate, and bivariate extension, with comparisons to existing operators.

Contribution

It develops a new class of operators using Polya distribution with Pochhammer $k$-symbol and studies their approximation properties and convergence behavior.

Findings

Operators converge uniformly to target functions.

Rate of convergence is explicitly estimated.

Bivariate generalization retains convergence properties.

Abstract

In our present investigation, we are concerned with the Kantorovich variant of Lupa\c{s}-Stancu operators based on Polya distribution with Pochhammer $k$-symbol. We briefly give some basic properties of the generalized operators and by making use of these results, we investigate convergence properties of the studied operators. Furthermore, the rate of convergence of these operators is obtained and Voronovskaja type theorem for the pointwise approximation is established. Then we construct bivariate generalization of the operators and we discuss some convergence properties. Finally, taking into account some illustrative graphics, we conclude our study with the comparison of the rate of convergence between our operators and other operators which are mentioned in the paper.