A New Aspect of q-Stancu Operators

In this thesis, the q-Stancu operators, which constitute a generalization of the q-Bernstein operators, a class of operators with significant importance in the literature, are examined comprehensively. Initially, the fundamental definitions and theorems used in operator theory are presented, followed by an explanation of the necessary concepts from q-analysis. The well-known basic properties of the q-Stancu operators, as defined by Nowak, are discussed, the relationships among these properties are analyzed, and relevant results from the literature are compiled. Subsequently, a new representation of these operators is obtained using the q-Pochhammer symbol. Through this new representation, some previously known properties are re-established via an alternative proof technique. Moreover, by employing certain relations arising from this new representation, a general recurrence formula for the m-th order moments of the q-Stancu operators is derived, and it is shown that these moments can be expressed in terms of lower-order moments. The study then defines the limit form of the q-Stancu operators and proves that these operators converge uniformly to the limit operator. Finally, the moments of the limit operator are computed, and the recurrence relations they satisfy are presented.