Exploring q-Stancu Operators via a New Representation

TL;DR

This paper introduces a new representation for q-Stancu operators, generalizing q-Bernstein operators, and establishes their properties, recurrence relations, and convergence behavior.

Contribution

It provides a novel q-Pochhammer based representation and analyzes the moments, limits, and convergence of q-Stancu operators.

Findings

Re-discovery of known properties using the new representation

Establishment of a recurrence relation for moments

Proof of uniform convergence of the operators

Abstract

This paper investigates the q-Stancu operators, which generalize the q-Bernstein operators, by developing a new representation in terms of the q-Pochhammer symbol. Based on this representation, some known properties are re-discovered, and a general recurrence relation for the moments is established. It is shown that higher-order moments can be expressed in terms of lower-order ones. Furthermore, the limit form of the operators is defined and their uniform convergence is proved. Finally, the moments of the limit operator and their recurrence relations are presented.