Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations

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 𝑞-difference calculus. Some of our results are new also in these particular discrete settings.


Introduction
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 1 and Diaz and Osler 2 , where basic approaches, definitions, and properties of the theory of fractional sums and differences were reported see also 3,4 . The cited papers discussed these notions on discrete sets formed by arithmetic or geometric sequences giving rise to fractional difference calculus or q-difference calculus . Recently, a series of papers continuing this research has appeared see, e.g., 5, 6 .
The extension of basic notions of fractional calculus to other discrete settings was performed in 7 , where fractional sums and differences have been introduced and studied in the framework of q, h -calculus, which can be reduced to ordinary difference calculus

Preliminaries
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 T. Before doing this, we recall the notion of Taylor monomials introduced in 9 . These monomials h n : T 2 → R, n ∈ N 0 are defined recursively as follows: Proof. This assertion can be proved by induction. If n 1, then 2.5 obviously holds. Let n ≥ 2 and assume that 2.5 holds with n replaced with n − 1, that is, a∇ −n 1 f t t a h n−2 t, ρ τ f τ ∇τ.

2.6
By the definition, the left-hand side of 2.5 is an antiderivative of a∇ −n 1 f t . We show that the right-hand side of 2.5 is an antiderivative of The formula 2.5 is a corner stone in the introduction of the nabla fractional integral a∇ −α f t for positive reals α. However, it requires a reasonable and natural extension of a discrete system of monomials h n , n ∈ N 0 to a continuous system h α , α ∈ R . This matter is closely related to a problem of an explicit form of h n . Of course, it holds h 1 t, s t − s for all t, s ∈ T. However, the calculation of h n for n > 1 is a difficult task which seems to be answerable only in some particular cases. It is well known that for T R, it holds respectively. In this connection, we recall a conventional notation used in ordinary difference calculus and q-calculus, namely, and j q j−1 r 0 q r q > 0 , n q ! n j 1 j q . To extend the meaning of these symbols also for noninteger values as it is required in the discrete fractional calculus , we recall some other necessary background of q-calculus. For any x ∈ R and 0 < q / 1, we set x q q x −1 / q −1 . By the continuity, we put x 1 x. Further, the q-Gamma function is defined for 0 < q < 1 as Note that this function satisfies the functional relation Γ q x 1 x q Γ q x and the condition Γ q 1 1. Using this, the q-binomial coefficient can be introduced as

2.13
Note that although the q-Gamma function is not defined at nonpositive integers, the formula 14 permits to calculate this ratio also at such the points. It is well known that if q → 1 − then Γ q x becomes the Euler Gamma function Γ x and analogously for the q-binomial coefficient . Among many interesting properties of the q-Gamma function and q-binomial coefficients, we mention q-Pascal rules Abstract and Applied Analysis 5 and the q-Vandermonde identity m j 0 x m − j q y j q q j 2 −mj xj x y m q , x,y ∈ R, m ∈ N 0 2.17 see 11 that turn out to be very useful in our further investigations. The computation of an explicit form of h n t, s can be performed also in a more general case. We consider here the time scale q,h . The forward and backward jump operator is the linear function σ t qt h and ρ t q −1 t − h , respectively. Similarly, the forward and backward graininess is given by μ t q − 1 t h and ν t q −1 μ t , respectively. In particular, if t 0 q h 1, then where the symbol σ i stands for the ith iterate of σ analogously, we use the symbol ρ i . To simplify the notation, we put q 1/q whenever considering the time scale T t 0 q,h or T σ i a q,h . Using the induction principle, we can verify that Taylor monomials on T t 0 q,h have the form Note that this result generalizes previous forms 2.10 and, moreover, enables its unified notation. In particular, if we introduce the symbolic q, h -power Now consider q, h -power 2.21 corresponding to the time scale and the formula 2.21 can be extended by These definitions are consistent, since it can be shown that Now the required extension of the monomial h n t, s corresponding to T t 0 q,h takes the form Another equivalent expression of h α t, s is provided by the following assertion.

Proposition 2.2.
Let α ∈ R, s, t ∈ T t 0 q,h and n ∈ N 0 be such that t σ n s . Then Proof. Let q > 1. Using the relations

2.32
The second equality in 2.30 follows from the identity 2.14 . The case q 1 results from 2.27 .
The key property of h α t, s follows from its differentiation. The symbol ∇ m q,h used in the following assertion and also undermentioned is the mth order nabla q, h -derivative on the time scale T t 0 q,h , defined for m 1 as and iteratively for higher orders.

2.35
The case m ≥ 2 can be verified by the induction principle.
We note that an extension of this property for derivatives of noninteger orders will be performed in Section 4. Now we can continue with the introduction of q, h -fractional integral and derivative of a function f : T a q,h → R. Let t ∈ T a q,h . Our previous considerations in particular, the Cauchy formula 2.5 along with the relations 2.22 and 2.29 warrant us to introduce the nabla q, h -fractional integral of order α ∈ R over the time scale interval a, t ∩ T a q,h as see also 7 . The nabla q, h -fractional derivative of order α ∈ R is then defined by where m ∈ Z is given by m − 1 < α ≤ m. For the sake of completeness, we put a∇ 0 q,h f t f t .

2.38
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.

A Linear Initial Value Problem
In this section, we are going to discuss the linear initial value problem and y α−j j 1, . . . , m are arbitrary real scalars.
If α is a positive integer, then 3.1 -3.2 becomes the standard discrete initial value problem. If α is not an integer, then applying the definition of nabla q, h -fractional derivatives, we can observe that 3.1 is of the general form , n being such that t σ n a ,  where

3.9
To calculate det H m , we employ some

3.11
Then we apply repeatedly this procedure to obtain the triangular matrix

3.12
Since h i−α σ k a , σ k−1 a ν σ k a i−α i 0, 1, . . . , m − 1 , we get Thus the matrix R m is regular, hence the corresponding mapping 3.6 is one to one. Now we approach a problem of the existence and uniqueness of 3.1 -3.2 . First we recall the general notion of ν-regressivity of a matrix function and a corresponding linear nabla dynamic system see 9 .
3.14 where I is the identity matrix. Further, we say that the linear dynamic system Considering a higher order linear difference equation, the notion of ν-regressivity for such an equation can be introduced by means of its transformation to the corresponding first order linear dynamic system. We are going to follow this approach and generalize the notion of ν-regressivity for the linear fractional difference equation 3.1 .

Definition 3.3.
Let α ∈ R and m ∈ Z be such that m − 1 < α ≤ m. Then 3.1 is called ν-regressive provided the matrix

3.17
If α is a positive integer, then both these introductions agree with the definition of ν-regressivity of a higher order linear difference equation presented in 9 .

3.21
and A t is given by 3.16 . The ν-regressivity of the matrix A t enables us to write hence, using the value of z σ m a , we can solve this system by the step method starting from t σ m 1 a . The solution y t of the original initial value problem 3.1 -3.2 is then given by the formula 3.19 .
Remark 3.6. 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 3.1 . We start with the following notion. Definition 3.7. Let γ ∈ R, 0 ≤ γ < 1. For m functions y j : T a q,h → R j 1, 2, . . . , m , we define the γ-Wronskian W γ y 1 , . . . , y m t as determinant of the matrix

3.23
3.25 The linearity of 3.1 implies that u t has to be its solution. Moreover, it holds

Two-Term Equation and q, h -Mittag-Leffler Function
Our main interest in this section is to find eigenfunctions of the fractional operator a∇ α q,h , α ∈ R . In other words, we wish to solve 3.1 in a special form Throughout this section, we assume that ν-regressivity condition for 4.1 is ensured, that is, 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 12-14 , where a discrete analogue of the Laplace transform turns out to be the most developed method. In this section, we describe the technique not utilizing the transform method, but directly originating from the role which is played by the Mittag-Leffler function in the continuous fractional calculus see, e.g., 15 . In particular, we introduce the notion of a discrete Mittag-Leffler function in a setting formed by the time scale T a q,h and demonstrate its significance with respect to eigenfunctions of the operator a∇ α q,h . These results generalize and extend those derived in 16, 17 . We start with the power rule stated in Lemma 2.3 and perform its extension to fractional integrals and derivatives. Proof. Let t ∈ T σ a q,h be such that t σ n a for some n ∈ Z . We have where we have used 2.30 on the second line and 2.17 on the last line.

4.5
Proof. Then the statement is an immediate consequence of Lemma 2.3.
Now we are in a position to introduce a q, h -discrete analogue of the Mittag-Leffler function. We recall that this function is essentially a generalized exponential function, and its two-parameter form more convenient in the fractional calculus can be introduced for T R by the series expansion The fractional calculus frequently employs 4.7 , because the function Considering the discrete calculus, the form 4.8 seems to be much more convenient for discrete extensions than the form 4.7 , which requires, among others, the validity of the law of exponents. The following introduction extends the discrete Mittag-Leffler function defined and studied in 20 for the case q h 1. Definition 4.3. Let α, β, λ ∈ R. We introduce the q, h -Mittag-Leffler function E s,λ α,β t by the series expansion It is easy to check that the series on the right-hand side converges absolutely if |λ| ν t α < 1.
As it might be expected, the particular q, h -Mittag-Leffler function where n ∈ Z satisfies t σ n a , is a solution of the equation that is, it is a discrete q, h -analogue of the exponential function.
The main properties of the q, h -Mittag-Leffler function are described by the following assertion. where n is a positive integer given by the condition |λ|ν σ n a α < 1. By Theorem 4.7, its solution can be expressed as a linear combination y t c 1 E a,λ α,α−1 t c 2 E a,λ α,α t .

4.33
Similarly we can determine y t for other choices of q and h. For comparative reasons, Figure 1 depicts in addition to the above case q h 1 the solution y t under particular choices q 1.2, h 0 the pure q-calculus , q 1, h 0.1 the pure h-calculus and also the solution of the corresponding continuous differential initial value problem.