This paper investigates 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

The fractional calculus is a research field of mathematical analysis which may be taken for an old as well as a modern topic. It is an old topic because of its long history starting from some notes and ideas of G. W. Leibniz and L. Euler. On the other hand, it is a modern topic due to its enormous development during the last two decades. The present interest of many scientists and engineers in the theory of fractional calculus has been initiated by applications of this theory as well as by new mathematical challenges.

The theory of discrete fractional calculus belongs among these challenges. Foundations of this theory were formulated in pioneering works by Agarwal [

The extension of basic notions of fractional calculus to other discrete settings was performed in [

The aim of this paper is to introduce some linear nabla

The structure of the paper is the following: Section

The basic definitions of fractional calculus on continuous or discrete settings usually originate from the Cauchy formula for repeated integration or summation, respectively. We state here its general form valid for arbitrary time scale

Let

This assertion can be proved by induction. If

The formula (

The computation of an explicit form of

Let

Using the induction principle, we can verify that Taylor monomials on

Discussing a reasonable generalization of

Another (equivalent) expression of

Let

Let

The key property of

Let

First let

We note that an extension of this property for derivatives of noninteger orders will be performed in Section

Now we can continue with the introduction of

As we noted earlier, a reasonable introduction of fractional integrals and fractional derivatives on arbitrary time scales remains an open problem. In the previous part, we have consistently used (and in the sequel, we shall consistently use) the time scale notation of main procedures and operations to outline a possible way out to further generalizations.

In this section, we are going to discuss the linear initial value problem

If

Let

The case

Now we approach a problem of the existence and uniqueness of (

An

Considering a higher order linear difference equation, the notion of

Let

The explicit expression of the

Let (

The conditions (

The previous assertion on the existence and uniqueness of the solution can be easily extended to the initial value problem involving nonhomogeneous linear equations as well as some nonlinear equations.

The final goal of this section is to investigate the structure of the solutions of (

Let

Note that the first row of this matrix involves fractional order integrals. It is a consequence of the form of initial conditions utilized in our investigations. Of course, this introduction of

Let functions

Let

The formula (

Our main interest in this section is to find eigenfunctions of the fractional operator

Discussions on methods of solving fractional difference equations are just at the beginning. Some techniques how to explicitly solve these equations (at least in particular cases) are exhibited, for example, in [

We start with the power rule stated in Lemma

Let

Let

Let

Proposition

Now we are in a position to introduce a

Considering the discrete calculus, the form (

Let

It is easy to check that the series on the right-hand side converges (absolutely) if

The main properties of the

(i) Let

(ii) Let

The part (i) follows immediately from Proposition

If

Let

The assertion follows from Theorem

Our final aim is to show that any solution of (

Let

The case

Now we summarize the results of Theorem

Let

We conclude this paper by the illustrating example.

Consider the initial value problem

Now we make a particular choice of the parameters

The research was supported by the research plan MSM 0021630518 “Simulation modelling of mechatronic systems” of the Ministry of Education, Youth and Sports of the Czech Republic, by Grant P201/11/0768 of the Czech Grant Agency and by Grant FSI-J-10-55 of the FME, Brno University of Technology.