Discrete Mittag-Leffler functions in linear fractional difference equations. (English) Zbl 1220.39010

Summary: We investigate some initial value problems in discrete fractional calculus. We introduce a linear difference equation of fractional order along with suitable initial conditions of fractional type and prove the existence and uniqueness of the solution. Then the structure of the solutions space is discussed, and, in a particular case, an explicit form of the general solution involving discrete analogues of Mittag-Leffler functions is presented. All our observations are performed on a special time scale which unifies and generalizes ordinary difference calculus and \(q\)-difference calculus. Some of our results are new also in these particular discrete settings.


39A13 Difference equations, scaling (\(q\)-differences)
39A06 Linear difference equations
26A33 Fractional derivatives and integrals
33E12 Mittag-Leffler functions and generalizations
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