A Note on q-Derivative Operator and Divided Difference
The q-analogue of the derivative operator is playing a more and more important role in mathematics and physics. Moreover, the divided difference, as an important and classical mathematical tool with a close relation to the derivative, is also used in many fields. In this paper, the connection between the q-derivative and the divided difference is investigated such that the q-derivative can be understood better. The q-derivative of higher order of a function f can be represented by the divided difference of f at some special nodes. Furthermore, the result is used to provide a new and easier proof of q-Leibnitz formula and its generalization.
Shaobo Zhong, Yimin Cheng and Xilong Qu
F. Cui "A Note on q-Derivative Operator and Divided Difference", Applied Mechanics and Materials, Vols. 50-51, pp. 377-381, 2011