This study defines a Hadamard fractional sum by use of the time-scale theory. Then a $h$-fractional difference is given and fundamental theorems are proved. Initial value problems of fractional difference equations are presented and their equivalent fractional sum equations are provided. The discrete Mittag-Leffler function solutions of linear fractional difference equations are obtained. It can be concluded that the new discrete fractional calculus of Hadamard type is well defined.

中文翻译：

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时间尺度上的 HADAMARD 分数阶微积分

本研究利用时间尺度理论定义了 Hadamard 分数和。然后一个$H$给出了分数差并证明了基本定理。给出了分数阶差分方程的初值问题，并给出了它们的等价分数和方程。得到了线性分数差分方程的离散Mittag-Leffler函数解。可以得出结论，新的 Hadamard 型离散分数阶微积分定义良好。