We utilize a new definition for the fractional delta operator and prove that it is equivalent by translation to the more commonly used operator. By means of the convolution operation we demonstrate that this new operator is strongly connected to the positivity, monotonicity, and convexity of the functions on which it operates. We also analyze the case of compositions of discrete fractional operators. Finally, since the operator we study here is translationally related to the more commonly used discrete fractional operators, we are able to establish many new results for all types of discrete fractional differences, and we explicitly demonstrate that our results improve all known existing results in the literature.

中文翻译：

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使用卷积方法的非局部算子的转移原理：分数单调性和凸性

我们对分数 delta 运算符使用了一个新定义，并通过转换为更常用的运算符来证明它是等效的。通过卷积运算，我们证明了这个新算子与它所操作的函数的正性、单调性和凸性密切相关。我们还分析了离散分数运算符的组合情况。最后，由于我们在这里研究的算子与更常用的离散分数算子在平移上相关，我们能够为所有类型的离散分数差异建立许多新结果，并且我们明确证明我们的结果改进了所有已知的现有结果文学。