Bivariate generalization for operators involving Appostol-Genocchi polynomial

This article is primarily concerned with the bivariate generalization of operators involving a class of orthogonal polynomials called Apostol-Genocchi polynomials. The rate of convergence can be determined in terms of partial and total modulus of continuity as well as the order of approximation can be achieved by means of a Lipschitz-type function and Peetre’s K-functional. In addition, we put forth a conceptual extension known as the “generalized boolean sum (GBS)” for these bivariate operators, which aims to establish the degree of approximation for Bögel continuous functions. In this study, we utilize the Mathematica Software to present a series of graphical illustrations that effectively showcase the rate of convergence for the bivariate operators. The graphs indicate that, in the case of certain functions, the bivariate operators exhibits superior convergence when \(\alpha \) is less than \(\beta \). Based on our analysis and comparison of the error of approximation between the bivariate operators and the corresponding GBS operators, it can be deduced that the GBS operators exhibit a faster convergence towards the function.