Hadamard Fractional Calculus on Time Scales

This study defines a Hadamard fractional sum by use of the time-scale theory. Then a [Formula: see text]-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.