(p,q)-Generalization of Szasz-Mirakyan Operators

TL;DR

This paper introduces a new class of (p,q)-generalized Szasz-Mirakyan operators, providing their explicit moments, convergence properties, and approximation capabilities, extending previous q-operators and establishing foundational results in approximation theory.

Contribution

The paper develops (p,q)-generalized Szasz-Mirakyan operators, deriving explicit moments, convergence theorems, and approximation properties, thus broadening the scope of operator theory in approximation.

Findings

Operators converge uniformly on bounded and unbounded intervals.

Explicit formulas for moments up to order 4 are provided.

Voronovskaya theorem is established for the new operators.

Abstract

In this paper, we introduce new modifications of Szasz-Mirakyan operators based on (p,q)-integers. We first give a recurrence relation for the moments of new operators and present explicit formula for the moments and central moments up to order 4. Some approximation properties of new operators are explored: the uniform convergence over bounded and unbounded intervals is established, direct approximation properties of the operators in terms of the moduli of smoothness is obtained and Voronovskaya theorem is presented. For the particular case p=1, the previous results for q-Szasz-Mirakyan operators are captured.