Moment Estimations of new Sz\'asz-Mirakyan-Durrmeyer operators

TL;DR

This paper investigates the moments and convergence properties of a Durrmeyer variant of Szász-Mirakyan operators, demonstrating their favorable properties without restrictions on the parameter, and providing explicit moment formulas.

Contribution

It introduces and analyzes a Durrmeyer variant of Szász-Mirakyan operators, establishing moments and convergence without parameter restrictions, which was not previously known.

Findings

The Durrmeyer variant has good approximation properties.

No restrictions on parameter β are needed for convergence.

Explicit formulas for moments are derived using special functions.

Abstract

In [10] Jain introduced the modified form of the Sz\'asz-Mirakjan operator, based on certain parameter $0\le\beta<1$. Several modifications of the operators are available in the literature. Here we consider actual Durrmeyer variant of the operators due to [10]. It is observed here that the Durrmeyer variant has the nice properties and one need not to take any restriction on $\beta$ in order to get convergence. We establish moments using the Tricomi's confluent hypergeometric function and Stirling numbers of first kind, also estimate some direct results