Abstract

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.

1. 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,)-calculus, which can be reduced to ordinary difference calculus and 𝑞-difference calculus via the choice 𝑞==1 and =0, respectively. This extension follows recent trends in continuous and discrete analysis, characterized by a unification and generalization, and resulting into the origin and progressive development of the time scales theory (see [8, 9]). Discussing problems of fractional calculus, a question concerning the introduction of (Hilger) fractional derivative or integral on arbitrary time scale turns out to be a difficult matter. Although first attempts have been already performed (see, e.g., [10]), results obtained in this direction seem to be unsatisfactory.

The aim of this paper is to introduce some linear nabla (𝑞,)-fractional difference equations (i.e., equations involving difference operators of noninteger orders) and investigate their basic properties. Some particular results concerning this topic are already known, either for ordinary difference equations or 𝑞-difference equations of fractional order (some relevant references will be mentioned in Section 4). We wish to unify them and also present results which are new even also in these particular discrete settings.

The structure of the paper is the following: Section 2 presents a necessary mathematical background related to discrete fractional calculus. In particular, we are going to make some general remarks concerning fractional calculus on arbitrary time scales. In Section 3, we consider a linear nabla (𝑞,)-difference equation of noninteger order and discuss the question of the existence and uniqueness of the solution for the corresponding initial value problem, as well as the question of a general solution of this equation. In Section 4, we consider a particular case of the studied equation and describe the base of its solutions space by the use of eigenfunctions of the corresponding difference operator. We show that these eigenfunctions can be taken for discrete analogues of the Mittag-Leffler functions.

2. 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 𝑛𝕋2, 𝑛0 are defined recursively as follows: 0(𝑡,𝑠)=1𝑠,𝑡𝕋(2.1) and, given 𝑛 for 𝑛0, we have 𝑛+1(𝑡,𝑠)=𝑡𝑠𝑛(𝜏,𝑠)𝜏𝑠,𝑡𝕋.(2.2) Now let 𝑓𝕋 be -integrable on [𝑎,𝑏]𝕋, 𝑎,𝑏𝕋. We put 𝑎1𝑓(𝑡)=𝑡𝑎𝑓(𝜏)𝜏𝑡𝕋,𝑎𝑡𝑏(2.3) and define recursively 𝑎𝑛𝑓(𝑡)=𝑡𝑎𝑎𝑛+1𝑓(𝜏)𝜏(2.4) for 𝑛=2,3,. Then we have the following.

Proposition 2.1 (Nabla Cauchy formula). Let 𝑛+, 𝑎,𝑏𝕋 and let 𝑓𝕋 be -integrable on [𝑎,𝑏]𝕋. If 𝑡𝕋, 𝑎𝑡𝑏, then 𝑎𝑛𝑓(𝑡)=𝑡𝑎𝑛1(𝑡,𝜌(𝜏))𝑓(𝜏)𝜏.(2.5)

Proof. This assertion can be proved by induction. If 𝑛=1, then (2.5) obviously holds. Let 𝑛2 and assume that (2.5) holds with 𝑛 replaced with 𝑛1, that is, 𝑎𝑛+1𝑓(𝑡)=𝑡𝑎𝑛2(𝑡,𝜌(𝜏))𝑓(𝜏)𝜏.(2.6) By the definition, the left-hand side of (2.5) is an antiderivative of 𝑎𝑛+1𝑓(𝑡). We show that the right-hand side of (2.5) is an antiderivative of 𝑡𝑎𝑛2(𝑡,𝜌(𝜏))𝑓(𝜏)𝜏. Indeed, it holds 𝑡𝑎𝑛1(𝑡,𝜌(𝜏))𝑓(𝜏)𝜏=𝑡𝑎𝑛1(𝑡,𝜌(𝜏))𝑓(𝜏)𝜏=𝑡𝑎𝑛2(𝑡,𝜌(𝜏))𝑓(𝜏)𝜏,(2.7) where we have employed the property 𝑡𝑎𝑔(𝑡,𝜏)𝜏=𝑡𝑎𝑔(𝑡,𝜏)𝜏+𝑔(𝜌(𝑡),𝑡)(2.8) (see [9, page 139]). Consequently, the relation (2.5) holds up to a possible additive constant. Substituting 𝑡=𝑎, we can find this additive constant zero.

The formula (2.5) is a corner stone in the introduction of the nabla fractional integral 𝑎𝛼𝑓(𝑡) for positive reals 𝛼. However, it requires a reasonable and natural extension of a discrete system of monomials (𝑛,𝑛0) to a continuous system (𝛼,𝛼+). This matter is closely related to a problem of an explicit form of 𝑛. Of course, it holds 1(𝑡,𝑠)=𝑡𝑠 for all 𝑡,𝑠𝕋. However, the calculation of 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=, it holds 𝑛(𝑡,𝑠)=(𝑡𝑠)𝑛𝑛!,(2.9) while for discrete time scales 𝕋= and 𝕋=𝑞={𝑞𝑘,𝑘}{0}, 𝑞>1, we have 𝑛(𝑡,𝑠)=𝑛1𝑗=0(𝑡𝑠+𝑗),𝑛!𝑛(𝑡,𝑠)=𝑛1𝑗=0𝑞𝑗𝑡𝑠𝑗𝑟=0𝑞𝑟,(2.10) respectively. In this connection, we recall a conventional notation used in ordinary difference calculus and 𝑞-calculus, namely,(𝑡𝑠)(𝑛)=𝑛1𝑗=0(𝑡𝑠+𝑗),(𝑡𝑠)(𝑛)̃𝑞=𝑡𝑛𝑛1𝑗=01̃𝑞𝑗𝑠𝑡(0<̃𝑞<1)(2.11) and [𝑗]𝑞=𝑗1𝑟=0𝑞𝑟(𝑞>0), [𝑛]𝑞!=𝑛𝑗=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 and 0<𝑞1, we set [𝑥]𝑞=(𝑞𝑥1)/(𝑞1). By the continuity, we put [𝑥]1=𝑥. Further, the 𝑞-Gamma function is defined for 0<̃𝑞<1 as Γ̃𝑞(𝑥)=(̃𝑞,̃𝑞)(1̃𝑞)1𝑥(̃𝑞𝑥,̃𝑞),(2.12) where (𝑝,̃𝑞)=𝑗=0(1𝑝̃𝑞𝑗), 𝑥{0,1,2,}. Note that this function satisfies the functional relation Γ̃𝑞(𝑥+1)=[𝑥]̃𝑞Γ̃𝑞(𝑥) and the condition Γ̃𝑞(1)=1. Using this, the 𝑞-binomial coefficient can be introduced as 𝑥𝑘̃𝑞=Γ̃𝑞(𝑥+1)Γ̃𝑞(𝑘+1)Γ̃𝑞(𝑥𝑘+1),𝑥,𝑘.(2.13) Note that although the 𝑞-Gamma function is not defined at nonpositive integers, the formula Γ̃𝑞(𝑥+𝑚)Γ̃𝑞(𝑥)=(1)𝑚̃𝑞𝑚2𝑥𝑚+Γ̃𝑞(1𝑥)Γ̃𝑞(1𝑥𝑚),𝑥,𝑚+(2.14) permits to calculate this ratio also at such the points. It is well known that if ̃𝑞1 then Γ̃𝑞(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 rules𝑥𝑘̃𝑞=𝑥1𝑘1̃𝑞+̃𝑞𝑘𝑘𝑥1̃𝑞𝑥𝑘,𝑥,𝑘,(2.15)̃𝑞=̃𝑞𝑥𝑘𝑥1𝑘1̃𝑞+𝑘𝑥1̃𝑞,𝑥,𝑘(2.16) and the 𝑞-Vandermonde identity 𝑚𝑗=0𝑥𝑚𝑗̃𝑞𝑦𝑗̃𝑞̃𝑞𝑗2𝑚𝑗+𝑥𝑗=𝑚𝑥+𝑦̃𝑞,𝑥,𝑦,𝑚0(2.17) (see [11]) that turn out to be very useful in our further investigations.

The computation of an explicit form of 𝑛(𝑡,𝑠) can be performed also in a more general case. We consider here the time scale 𝕋𝑡0(𝑞,)=𝑡0𝑞𝑘+[𝑘]𝑞,𝑘1𝑞,𝑡0>0,𝑞1,0,𝑞+>1(2.18) (see also [7]). Note that if 𝑞=1 then the cluster point /(1𝑞)= is not involved in 𝕋𝑡0(𝑞,). The forward and backward jump operator is the linear function 𝜎(𝑡)=𝑞𝑡+ and 𝜌(𝑡)=𝑞1(𝑡), respectively. Similarly, the forward and backward graininess is given by 𝜇(𝑡)=(𝑞1)𝑡+ and 𝜈(𝑡)=𝑞1𝜇(𝑡), respectively. In particular, if 𝑡0=𝑞==1, then 𝕋𝑡0(𝑞,) becomes , and if 𝑡0=1, 𝑞>1, =0, then 𝕋𝑡0(𝑞,) is reduced to 𝑞.

Let 𝑎𝕋𝑡0(𝑞,), 𝑎>/(1𝑞) be fixed. Then we introduce restrictions of the time scale 𝕋𝑡0(𝑞,) by the relation 𝕋𝜎𝑖(𝑎)(𝑞,)=𝑡𝕋𝑡0(𝑞,),𝑡𝜎𝑖(𝑎),𝑖=0,1,,(2.19) where the symbol 𝜎𝑖 stands for the 𝑖th iterate of 𝜎 (analogously, we use the symbol 𝜌𝑖). To simplify the notation, we put ̃𝑞=1/𝑞 whenever considering the time scale 𝕋𝑡0(𝑞,) or 𝕋𝜎𝑖(𝑎)(𝑞,).

Using the induction principle, we can verify that Taylor monomials on 𝕋𝑡0(𝑞,) have the form 𝑛(𝑡,𝑠)=𝑛1𝑗=0𝜎𝑗(𝑡)𝑠[𝑛]𝑞!=𝑛1𝑗=0𝑡𝜌𝑗(𝑠)[𝑛]̃𝑞!.(2.20) Note that this result generalizes previous forms (2.10) and, moreover, enables its unified notation. In particular, if we introduce the symbolic (𝑞,)-power(𝑡𝑠)((𝑛)̃𝑞,)=𝑛1𝑗=0𝑡𝜌𝑗(𝑠)(2.21) unifying (2.11), then the Cauchy formula (2.5) can be rewritten for 𝕋=𝕋𝑡0(𝑞,) as 𝑎𝑛𝑓(𝑡)=𝑡𝑎(𝑡𝜌(𝜏))(𝑛1)̃𝑞,[]𝑛1̃𝑞!𝑓(𝜏)𝜏.(2.22)

Discussing a reasonable generalization of (𝑞,)-power (2.21) to real values 𝛼 instead of integers 𝑛, we recall broadly accepted extensions of its particular cases (2.11) in the form (𝑡𝑠)(𝛼)=Γ(𝑡𝑠+𝛼)Γ(𝑡𝑠),(𝑡𝑠)(𝛼)̃𝑞=𝑡𝛼(𝑠/𝑡,̃𝑞)(̃𝑞𝛼𝑠/𝑡,̃𝑞),𝑡0.(2.23) Now, we assume 𝑠,𝑡𝕋𝑡0(𝑞,), 𝑡𝑠>/(1𝑞). First, consider (𝑞,)-power (2.21) corresponding to the time scale 𝕋𝑡0(𝑞,), where 𝑞>1. Then we can rewrite (2.21) as (𝑡𝑠)(𝑛)̃𝑞,=𝑡+̃𝑞1̃𝑞𝑛𝑛1𝑗=01̃𝑞𝑗𝑠+̃𝑞/(1̃𝑞)[𝑡][𝑠])𝑡+̃𝑞/(1̃𝑞)=((𝑛)̃𝑞,(2.24) where [𝑡]=𝑡+̃𝑞/(1̃𝑞) and [𝑠]=𝑠+̃𝑞/(1̃𝑞). A required extension of (𝑞,)-power (2.21) is then provided by the formula (𝑡𝑠)((𝛼)̃𝑞,)[𝑡][𝑠])=((𝛼)̃𝑞.(2.25) Now consider (𝑞,)-power (2.21) corresponding to the time scale 𝕋𝑡0(𝑞,), where 𝑞=1. Then (𝑡𝑠)(𝑛)(1,)=𝑛1𝑗=0(𝑡𝑠+𝑗)=𝑛𝑛1𝑗=0𝑡𝑠+𝑗=𝑛((𝑡𝑠)/+𝑛1)!((𝑡𝑠)/1)!(2.26) and the formula (2.21) can be extended by (𝑡𝑠)(𝛼)(1,)=𝛼Γ((𝑡𝑠)/+𝛼)Γ((𝑡𝑠)/).(2.27) These definitions are consistent, since it can be shown that lim̃𝑞1([𝑡][𝑠])(𝛼)̃𝑞=(𝑡𝑠)(𝛼)(1,).(2.28) Now the required extension of the monomial 𝑛(𝑡,𝑠) corresponding to 𝕋𝑡0(𝑞,) takes the form 𝛼(𝑡,𝑠)=(𝑡𝑠)((𝛼)̃𝑞,)Γ̃𝑞(𝛼+1).(2.29)

Another (equivalent) expression of 𝛼(𝑡,𝑠) is provided by the following assertion.

Proposition 2.2. Let 𝛼, 𝑠,𝑡𝕋𝑡0(𝑞,) and 𝑛0 be such that 𝑡=𝜎𝑛(𝑠). Then 𝛼(𝑡,𝑠)=(𝜈(𝑡))𝛼𝛼+𝑛1𝑛1̃𝑞=(𝜈(𝑡))𝛼𝛼1𝑛1̃𝑞(1)𝑛1̃𝑞𝑛2𝛼(𝑛1)+.(2.30)

Proof. Let 𝑞>1. Using the relations [𝑡]=𝜈(𝑡)(,[𝑠]1̃𝑞)[𝑡]=̃𝑞𝑛,(2.31) we can derive that 𝛼[𝑡](𝑡,𝑠)=𝛼([𝑠]/[𝑡]),̃𝑞Γ̃𝑞(𝛼+1)(̃𝑞𝛼[𝑠]/[𝑡],̃𝑞)=(1̃𝑞)𝛼𝜈(𝑡)𝛼(̃𝑞𝑛),̃𝑞Γ̃𝑞(𝛼+1)̃𝑞𝛼+𝑛,̃𝑞=(𝜈(𝑡))𝛼Γ̃𝑞(𝛼+𝑛)Γ̃𝑞(𝛼+1)Γ̃𝑞(𝑛)=(𝜈(𝑡))𝛼𝛼+𝑛1𝑛1̃𝑞.(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 𝛼(𝑡,𝑠) follows from its differentiation. The symbol 𝑚(𝑞,) used in the following assertion (and also undermentioned) is the 𝑚th order nabla (𝑞,)-derivative on the time scale 𝕋𝑡0(𝑞,), defined for 𝑚=1 as (𝑞,)𝑓(𝑡)=𝑓(𝑡)𝑓(𝜌(𝑡))=𝜈(𝑡)𝑓(𝑡)𝑓(̃𝑞(𝑡))(1̃𝑞)𝑡+̃𝑞(2.33) and iteratively for higher orders.

Lemma 2.3. Let 𝑚+, 𝛼, 𝑠,𝑡𝕋𝑡0(𝑞,) and 𝑛+, 𝑛𝑚 be such that 𝑡=𝜎𝑛(𝑠). Then 𝑚(𝑞,)𝛼(𝑡,𝑠)=𝛼𝑚(𝑡,𝑠),𝛼{0,1,,𝑚1},0,𝛼{0,1,,𝑚1}.(2.34)

Proof. First let 𝑚=1. For 𝛼=0 we get 0(𝑡,𝑠)=1 and the first nabla (𝑞,)-derivative is zero. If 𝛼0, then by (2.30) and (2.16), we have (𝑞,)𝛼(𝑡,𝑠)=𝛼(𝑡,𝑠)𝛼(𝜌(𝑡),𝑠)=1𝜈(𝑡)𝜈(𝑡)(𝜈(𝑡))𝛼𝛼+𝑛1𝑛1̃𝑞(𝜈(𝜌(𝑡)))𝛼𝛼+𝑛2𝑛2̃𝑞=(𝜈(𝑡))𝛼1𝛼+𝑛1𝑛1̃𝑞̃𝑞𝛼𝛼+𝑛2𝑛2̃𝑞=𝛼1(𝑡,𝑠).(2.35) The case 𝑚2 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,)-fractional integral and derivative of a function 𝕋𝑓𝑎(𝑞,). Let 𝕋𝑡𝑎(𝑞,). 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,)-fractional integral of order 𝛼+ over the time scale interval 𝕋[𝑎,𝑡]𝑎(𝑞,) as 𝑎𝛼(𝑞,)𝑓(𝑡)=𝑡𝑎𝛼1(𝑡,𝜌(𝜏))𝑓(𝜏)𝜏(2.36) (see also [7]). The nabla (𝑞,)-fractional derivative of order 𝛼+ is then defined by 𝑎𝛼(𝑞,)𝑓(𝑡)=𝑚𝑎(𝑞,)(𝑚𝛼)(𝑞,)𝑓(𝑡),(2.37) where 𝑚+ is given by 𝑚1<𝛼𝑚. For the sake of completeness, we put 𝑎0(𝑞,)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.

3. A Linear Initial Value Problem

In this section, we are going to discuss the linear initial value problem 𝑚𝑗=1𝑝𝑚𝑗+1(𝑡)𝑎𝛼𝑗+1(𝑞,)𝑦(𝑡)+𝑝0𝕋(𝑡)𝑦(𝑡)=0,𝑡𝜎𝑚+1(𝑎)(𝑞,),(3.1)𝑎𝛼𝑗(𝑞,)|||𝑦(𝑡)𝑡=𝜎𝑚(𝑎)=𝑦𝛼𝑗,𝑗=1,2,,𝑚,(3.2) where 𝛼+ and 𝑚+ are such that 𝑚1<𝛼𝑚. Further, we assume that 𝑝𝑗(𝑡) are arbitrary real-valued functions on 𝕋𝜎𝑚+1(𝑎)(𝑞,)(𝑗=1,,𝑚1), 𝑝𝑚(𝑡)=1 on 𝕋𝜎𝑚+1(𝑎)(𝑞,) and 𝑦𝛼𝑗(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,)-fractional derivatives, we can observe that (3.1) is of the general form 𝑛1𝑖=0𝑎𝑖𝜌(𝑡)𝑦𝑖𝕋(𝑡)=0,𝑡𝜎𝑚+1(𝑎)(𝑞,),𝑛beingsuchthat𝑡=𝜎𝑛(𝑎),(3.3) which is usually referred to as the equation of Volterra type. If such an equation has two different solutions, then their values differ at least at one of the points 𝜎(𝑎),𝜎2(𝑎),,𝜎𝑚(𝑎). In particular, if 𝑎0(𝑡)0 for all 𝕋𝑡𝜎𝑚+1(𝑎)(𝑞,), then arbitrary values of 𝑦(𝜎(𝑎)),𝑦(𝜎2(𝑎)),,𝑦(𝜎𝑚(𝑎)) determine uniquely the solution 𝑦(𝑡) for all 𝕋𝑡𝜎𝑚+1(𝑎)(𝑞,). We show that the values 𝑦𝛼1,𝑦𝛼2,,𝑦𝛼𝑚, introduced by (3.2), keep the same properties.

Proposition 3.1. Let 𝕋𝑦𝜎(𝑎)(𝑞,) be a function. Then (3.2) represents a one-to-one mapping between the vectors (𝑦(𝜎(𝑎)),𝑦(𝜎2(𝑎)),,𝑦(𝜎𝑚(𝑎))) and (𝑦𝛼1,𝑦𝛼2,,𝑦𝛼𝑚).

Proof. The case 𝛼+ is well known from the literature. Let 𝛼+. We wish to show that the values of 𝑦(𝜎(𝑎)),𝑦(𝜎2(𝑎)),,𝑦(𝜎𝑚(𝑎)) determine uniquely the values of 𝑎𝛼1(𝑞,)||𝑦(𝑡)𝑡=𝜎𝑚(𝑎),𝑎𝛼2(𝑞,)||𝑦(𝑡)𝑡=𝜎𝑚(𝑎),,𝑎𝛼𝑚(𝑞,)||𝑦(𝑡)𝑡=𝜎𝑚(𝑎)(3.4) and vice versa. Utilizing the relation 𝑎𝛼𝑗(𝑞,)|||𝑦(𝑡)𝑡=𝜎𝑚(𝑎)=𝑚𝑘=1𝜈𝜎𝑚𝑘+1(𝑎)𝑗1𝛼𝜎𝑚(𝑎),𝜎𝑚𝑘𝑦𝜎(𝑎)𝑚𝑘+1(𝑎)(3.5) (see [7, Propositions  1 and  3] with respect to (2.30)), we can rewrite (3.2) as the linear mapping 𝑚𝑘=1𝑟𝑗𝑘𝑦𝜎𝑚𝑘+1(𝑎)=𝑦𝛼𝑗,𝑗=1,,𝑚,(3.6) where 𝑟𝑗𝑘𝜎=𝜈𝑚𝑘+1(𝑎)𝑗1𝛼𝜎𝑚(𝑎),𝜎𝑚𝑘(𝑎),𝑗,𝑘=1,,𝑚(3.7) are elements of the transformation matrix 𝑅𝑚. We show that 𝑅𝑚 is regular. Obviously, det𝑅𝑚=𝑚𝑘=1𝜈𝜎𝑘(𝑎)det𝐻𝑚,(3.8) where 𝐻𝑚=𝛼𝜎𝑚(𝑎),𝜎𝑚1(𝑎)𝛼𝜎𝑚(𝑎),𝜎𝑚2(𝑎)𝛼(𝜎𝑚(𝑎),𝑎)1𝛼𝜎𝑚(𝑎),𝜎𝑚1(𝑎)1𝛼𝜎𝑚(𝑎),𝜎𝑚2(𝑎)1𝛼(𝜎𝑚(𝑎),𝑎)𝑚1𝛼𝜎𝑚(𝑎),𝜎𝑚1(𝑎)𝑚1𝛼𝜎𝑚(𝑎),𝜎𝑚2(𝑎)𝑚1𝛼(𝜎𝑚(𝑎),𝑎).(3.9) To calculate det𝐻𝑚, we employ some elementary operations preserving the value of det𝐻𝑚. Using the properties 𝑖𝛼𝜎𝑚(𝑎),𝜎(𝑎)𝜈(𝜎𝑚(𝑎))𝑖𝛼1𝜎𝑚(𝑎),𝜎=(𝑎)𝑖𝛼𝜎𝑚1(𝑎),𝜎(𝑎)(𝑖=1,2,,𝑚1,𝑙=0,1,𝑚2),𝑖𝛼𝜎𝑚(𝑎),𝜎𝑚1(𝑎)𝜈(𝜎𝑚(𝑎))𝑖𝛼1𝜎𝑚(𝑎),𝜎𝑚1(𝑎)=0,(3.10) which follow from Lemma 2.3, we multiply the 𝑖th row (𝑖=1,2,,𝑚1) of 𝐻𝑚 by 𝜈(𝜎𝑚(𝑎)) and add it to the successive one. We arrive at the form 𝛼𝜎𝑚(𝑎),𝜎𝑚100(𝑎)𝛼𝜎𝑚(𝑎),𝜎𝑚2(𝑎)𝛼(𝜎𝑚(𝑎),𝑎)𝐻𝑚1.(3.11) Then we apply repeatedly this procedure to obtain the triangular matrix 𝛼𝜎𝑚(𝑎),𝜎𝑚1(𝑎)𝛼𝜎𝑚(𝑎),𝜎𝑚2(𝑎)𝛼(𝜎𝑚0(𝑎),𝑎)1𝛼𝜎𝑚1(𝑎),𝜎𝑚2(𝑎)1𝛼𝜎𝑚1(𝑎),𝑎00𝑚1𝛼(𝜎(𝑎),𝑎).(3.12) Since 𝑖𝛼(𝜎𝑘(𝑎),𝜎𝑘1(𝑎))=(𝜈(𝜎𝑘(𝑎))𝑖𝛼(𝑖=0,1,,𝑚1), we get det𝐻𝑚=𝑚𝑘=1𝜈𝜎𝑘(𝑎)𝑚𝑘𝛼,thatis,det𝑅𝑚=𝑚𝑘=1𝜈𝜎𝑘(𝑎)𝑚𝑘𝛼+10.(3.13) Thus the matrix 𝑅𝑚 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]).

Definition 3.2. An 𝑛×𝑛-matrix-valued function 𝐴(𝑡) on a time scale 𝕋 is called 𝜈-regressive provided det(𝐼𝜈(𝑡)𝐴(𝑡))0𝑡𝕋𝜅,(3.14) where 𝐼 is the identity matrix. Further, we say that the linear dynamic system 𝑧(𝑡)=𝐴(𝑡)𝑧(𝑡)(3.15) is 𝜈-regressive provided that 𝐴(𝑡) is 𝜈-regressive.

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+ and 𝑚+ be such that 𝑚1<𝛼𝑚. Then (3.1) is called 𝜈-regressive provided the matrix 𝑝𝐴(𝑡)=0100001000010(𝑡)𝜈𝑚𝛼(𝑡)𝑝1(𝑡)𝑝𝑚2(𝑡)𝑝𝑚1(𝑡)(3.16) is 𝜈-regressive.

Remark 3.4. The explicit expression of the 𝜈-regressivity property for (3.1) can be read as 1+𝑚1𝑗=1𝑝𝑚𝑗(𝑡)(𝜈(𝑡))𝑗+𝑝0(𝑡)(𝜈(𝑡))𝛼𝕋0𝑡𝜎𝑚+1(𝑎)(𝑞,).(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].

Theorem 3.5. Let (3.1) be 𝜈-regressive. Then the problem (3.1)-(3.2) has a unique solution defined for all 𝕋𝑡𝜎(𝑎)(𝑞,).

Proof. The conditions (3.2) enable us to determine the values of 𝑦(𝜎(𝑎)),𝑦(𝜎2(𝑎)),,𝑦(𝜎𝑚(𝑎)) by the use of (3.6). To calculate the values of 𝑦(𝜎𝑚+1(𝑎)),𝑦(𝜎𝑚+2(𝑎)),, we perform the transformation 𝑧𝑗(𝑡)=𝑎𝛼𝑚+𝑗1(𝑞,)𝕋𝑦(𝑡),𝑡𝜎𝑗(𝑎)(𝑞,),𝑗=1,2,,𝑚(3.18) which allows us to rewrite (3.1) into a matrix form. Before doing this, we need to express 𝑦(𝑡) in terms of 𝑧1(𝑡),𝑧1(𝜌(𝑡)),,𝑧1(𝜎(𝑎)). Applying the relation 𝑎𝑎𝑚𝛼(𝑞,)(𝑚𝛼)(𝑞,)𝑦(𝑡)=𝑦(𝑡) (see [7]) and expanding the fractional derivative, we arrive at 𝑦(𝑡)=𝑎𝑚𝛼(𝑞,)𝑧1𝑧(𝑡)=1(𝑡)𝜈𝑚𝛼+(𝑡)𝑎𝜌(𝑡)𝛼𝑚1(𝑡,𝜌(𝜏))𝑧1(𝜏)𝜏.(3.19) Therefore, the problem (3.1)-(3.2) can be rewritten to the vector form 𝑎(𝑞,)𝕋𝑧(𝑡)=𝐴(𝑡)𝑧(𝑡)+𝑏(𝑡),𝑡𝜎𝑚+1(𝑎)(𝑞,),𝑧(𝜎𝑚𝑦(𝑎))=𝛼𝑚,,𝑦𝛼1𝑇,(3.20) where 𝑧𝑧(𝑡)=1(𝑡),,𝑧𝑚(𝑡)𝑇,𝑏(𝑡)=0,,0,𝑝0(𝑡)𝑎𝜌(𝑡)𝛼𝑚1(𝑡,𝜌(𝜏))𝑧1(𝜏)𝜏𝑇(3.21) and 𝐴(𝑡) is given by (3.16). The 𝜈-regressivity of the matrix 𝐴(𝑡) enables us to write 𝑧(𝑡)=(𝐼𝜈(𝑡)𝐴(𝑡))1𝕋(𝑧(𝜌(𝑡))+𝜈(𝑡)𝑏(𝑡)),𝑡𝜎𝑚+1(𝑎)(𝑞,),(3.22) hence, using the value of 𝑧(𝜎𝑚(𝑎)), we can solve this system by the step method starting from 𝑡=𝜎𝑚+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, 0𝛾<1. For 𝑚 functions 𝑦𝑗𝕋𝑎(𝑞,)(𝑗=1,2,,𝑚), we define the 𝛾-Wronskian 𝑊𝛾(𝑦1,,𝑦𝑚)(𝑡) as determinant of the matrix 𝑉𝛾𝑦1,,𝑦𝑚(𝑡)=𝑎𝛾(𝑞,)𝑦1(𝑡)𝑎𝛾(𝑞,)𝑦2(𝑡)𝑎𝛾(𝑞,)𝑦𝑚(𝑡)𝑎1𝛾(𝑞,)𝑦1(𝑡)𝑎1𝛾(𝑞,)𝑦2(𝑡)𝑎1𝛾(𝑞,)𝑦𝑚(𝑡)𝑎𝑚1𝛾(𝑞,)𝑦1(𝑡)𝑎𝑚1𝛾(𝑞,)𝑦2(𝑡)𝑎𝑚1𝛾(𝑞,)𝑦𝑚𝕋(𝑡),𝑡𝜎𝑚(𝑎)(𝑞,).(3.23)

Remark 3.8. 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 𝑊𝛾(𝑦1,,𝑦𝑚)(𝑡) coincides for 𝛾=0 with the classical definition of the Wronskian (see [8]). Moreover, it holds 𝑊𝛾(𝑦1,,𝑦𝑚)(𝑡)=𝑊0(𝑎𝛾(𝑞,)𝑦1,,𝑎𝛾(𝑞,)𝑦𝑚)(𝑡).

Theorem 3.9. Let functions 𝑦1(𝑡),,𝑦𝑚(𝑡) be solutions of the 𝜈-regressive equation (3.1) and let 𝑊𝑚𝛼(𝑦1,,𝑦𝑚)(𝜎𝑚(𝑎))0. Then any solution 𝑦(𝑡) of (3.1) can be written in the form 𝑦(𝑡)=𝑚𝑘=1𝑐𝑘𝑦𝑘𝕋(𝑡),𝑡𝜎(𝑎)(𝑞,),(3.24) where 𝑐1,,𝑐𝑚 are real constants.

Proof. Let 𝑦(𝑡) be a solution of (3.1). By Proposition 3.1, there exist real scalars 𝑦𝛼1,,𝑦𝛼𝑚 such that 𝑦(𝑡) is satisfying (3.2). Now we consider the function 𝑢(𝑡)=𝑚𝑘=1𝑐𝑘𝑦𝑘(𝑡), where the 𝑚-tuple (𝑐1,,𝑐𝑚) is the unique solution of 𝑉𝑚𝛼𝑦1,,𝑦𝑚(𝜎𝑚𝑐(𝑎))1𝑐2𝑐𝑚=𝑦𝛼𝑚𝑦𝛼𝑚+1𝑦𝛼1.(3.25) The linearity of (3.1) implies that 𝑢(𝑡) has to be its solution. Moreover, it holds 𝑎𝛼𝑗(𝑞,)|||𝑢(𝑡)𝑡=𝜎𝑚(𝑎)=𝑦𝛼𝑗,𝑗=1,2,,𝑚,(3.26) hence 𝑢(𝑡) is a solution of the initial value problem (3.1)-(3.2). By Theorem 3.5, it must be 𝑦(𝑡)=𝑢(𝑡) for all 𝕋𝑡𝜎(𝑎)(𝑞,) and (3.24) holds.

Remark 3.10. The formula (3.24) is essentially an expression of the general solution of (3.1).

4. Two-Term Equation and (𝑞,)-Mittag-Leffler Function

Our main interest in this section is to find eigenfunctions of the fractional operator 𝑎𝛼(𝑞,), 𝛼+. In other words, we wish to solve (3.1) in a special form 𝑎𝛼(𝑞,)𝕋𝑦(𝑡)=𝜆𝑦(𝑡),𝜆,𝑡𝜎𝑚+1(𝑎)(𝑞,).(4.1) Throughout this section, we assume that 𝜈-regressivity condition for (4.1) is ensured, that is, 𝜆(𝜈(𝑡))𝛼1.(4.2)

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 [1214], 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 𝕋𝑎(𝑞,) and demonstrate its significance with respect to eigenfunctions of the operator 𝑎𝛼(𝑞,). 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.

Proposition 4.1. Let 𝛼+, 𝛽 and 𝕋𝑡𝜎(𝑎)(𝑞,). Then it holds 𝑎𝛼(𝑞,)𝛽(𝑡,𝑎)=𝛼+𝛽(𝑡,𝑎).(4.3)

Proof. Let 𝕋𝑡𝜎(𝑎)(𝑞,) be such that 𝑡=𝜎𝑛(𝑎) for some 𝑛+. We have 𝑎𝛼(𝑞,)𝛽(𝑡,𝑎)=𝑛1𝑘=0𝛼1𝑡,𝜌𝑘+1𝜈𝜌(𝑡)𝑘(𝑡)𝛽𝜌𝑘=(𝑡),𝑎𝑛1𝑘=0(𝜈(𝑡))𝛼1𝑘𝛼̃𝑞(1)𝑘̃𝑞2(𝛼1)𝑘+𝑘+1̃𝑞𝑘×𝜈𝜌𝜈(𝑡)𝑘(𝑡)𝛽𝛽1𝑛𝑘1̃𝑞(1)𝑛𝑘1̃𝑞2𝛽(𝑛𝑘1)+𝑛𝑘=(𝜈(𝑡))𝛼+𝛽𝑛1𝑘=0𝑘𝛼̃𝑞𝛽1𝑛𝑘1̃𝑞(1)𝑛1̃𝑞𝑘2𝑛2𝑘(𝑛1)+𝑘𝛼++𝛽(𝑛1)=(𝜈(𝑡))𝛼+𝛽𝑛1𝑘=0𝛼𝑛𝑘1̃𝑞𝑘𝛽1̃𝑞×(1)𝑛1̃𝑞(𝑛𝑘1)2𝑛2(𝑛𝑘1)(𝑛1)+(𝑛𝑘1)𝛼++𝛽(𝑛1)=(𝜈(𝑡))𝛼+𝛽𝑛1𝑘=0𝛼𝑛𝑘1̃𝑞𝑘𝛽1̃𝑞(1)𝑛1̃𝑞𝑘2𝑛2𝑘(𝑛1)𝑘𝛼+(𝛼+𝛽)(𝑛1)+=(𝜈(𝑡))𝛼+𝛽𝛼𝛽1𝑛1̃𝑞(1)𝑛1̃𝑞𝑛2(𝛼+𝛽)(𝑛1)+=𝛼+𝛽(𝑡,𝑎),(4.4) where we have used (2.30) on the second line and (2.17) on the last line.

Corollary 4.2. Let 𝛼+, 𝛽, 𝕋𝑡𝜎𝑚+1(𝑎)(𝑞,), where 𝑚+ is satisfying 𝑚1<𝛼𝑚. Then 𝑎𝛼(𝑞,)𝛽(𝑡,𝑎)=𝛽𝛼(𝑡,𝑎),𝛽𝛼{1,,𝑚},0,𝛽𝛼{1,,𝑚}.(4.5)

Proof. Proposition 4.1 implies that 𝑎𝛼(𝑞,)𝛽(𝑡,𝑎)=𝑚(𝑞,)𝑎(𝑚𝛼)(𝑞,)𝛽(𝑡,𝑎)=𝑚(𝑞,)𝑚+𝛽𝛼(𝑡,𝑎).(4.6) Then the statement is an immediate consequence of Lemma 2.3.

Now we are in a position to introduce a (𝑞,)-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= by the series expansion 𝐸𝛼,𝛽(𝑡)=𝑘=0𝑡𝑘Γ(𝛼𝑘+𝛽),𝛼,𝛽+,𝑡.(4.7) The fractional calculus frequently employs (4.7), because the function 𝑡𝛽1𝐸𝛼,𝛽(𝜆𝑡𝛼)=𝑘=0𝜆𝑘𝑡𝛼𝑘+𝛽1Γ(𝛼𝑘+𝛽)(4.8) (a modified Mittag-Leffler function, see [15]) satisfies under special choices of 𝛽 a continuous (differential) analogy of (4.1). Some extensions of the definition formula (4.7) and their utilization in special fractional calculus operators can be found in [18, 19].

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==1.

Definition 4.3. Let 𝛼,𝛽,𝜆. We introduce the (𝑞,)-Mittag-Leffler function 𝐸𝑠,𝜆𝛼,𝛽(𝑡) by the series expansion 𝐸𝑠,𝜆𝛼,𝛽(𝑡)=𝑘=0𝜆𝑘𝛼𝑘+𝛽1=(𝑡,𝑠)𝑘=0𝜆𝑘(𝑡𝑠)((𝛼𝑘+𝛽1)̃𝑞,)Γ̃𝑞(𝕋𝛼𝑘+𝛽),𝑠,𝑡𝑎(𝑞,),𝑡𝑠.(4.9)

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,)-Mittag-Leffler function 𝐸𝑎,𝜆1,1(𝑡)=𝑛1𝑘=01𝜌1𝜆𝜈𝑘(𝑡),(4.10) where 𝑛+ satisfies 𝑡=𝜎𝑛(𝑎), is a solution of the equation (𝑞,)𝕋𝑦(𝑡)=𝜆𝑦(𝑡),𝑡𝜎(𝑎)(𝑞,),(4.11) that is, it is a discrete (𝑞,)-analogue of the exponential function.

The main properties of the (𝑞,)-Mittag-Leffler function are described by the following assertion.

Theorem 4.4. (i) Let 𝜂+ and 𝕋𝑡𝜎(𝑎)(𝑞,). Then 𝑎𝜂(𝑞,)𝐸𝑎,𝜆𝛼,𝛽(𝑡)=𝐸𝑎,𝜆𝛼,𝛽+𝜂(𝑡).(4.12)
(ii) Let 𝜂+, 𝑚+ be such that 𝑚1<𝜂𝑚 and let 𝛼𝑘+𝛽1{0,1,,𝑚+1} for all 𝑘+. If 𝕋𝑡𝜎𝑚+1(𝑎)(𝑞,), then 𝑎𝜂(𝑞,)𝐸𝑎,𝜆𝛼,𝛽𝐸(𝑡)=𝑎,𝜆𝛼,𝛽𝜂(𝑡),𝛽𝜂{0,1,,𝑚+1},𝜆𝐸𝑎,𝜆𝛼,𝛽𝜂+𝛼(𝑡),𝛽𝜂{0,1,,𝑚+1}.(4.13)

Proof. The part (i) follows immediately from Proposition 4.1. Considering the part (ii), we can write 𝑎𝜂(𝑞,)𝐸𝑎,𝜆𝛼,𝛽(𝑡)=𝑎𝜂(𝑞,)𝑘=0𝜆𝑘𝛼𝑘+𝛽1(𝑡,𝑎)=𝑘=0𝜆𝑘𝑎𝜂(𝑞,)𝛼𝑘+𝛽1(𝑡,𝑎)(4.14) due to the absolute convergence property.
If 𝑘+, then Corollary 4.2 implies the relation 𝑎𝜂(𝑞,)𝛼𝑘+𝛽1(𝑡,𝑎)=𝛼𝑘+𝛽𝜂1(𝑡,𝑎)(4.15) due to the assumption 𝛼𝑘+𝛽1{0,1,,𝑚+1}. If 𝑘=0, then two possibilities may occur. If 𝛽𝜂{0,1,,𝑚+1}, we get (4.15) with 𝑘=0 which implies the validity of (4.13)1. If 𝛽𝜂{0,1,,𝑚+1}, the nabla (𝑞,)-fractional derivative of this term is zero and by shifting the index 𝑘, we obtain (4.13)2.

Corollary 4.5. Let 𝛼+ and 𝑚+ be such that 𝑚1<𝛼𝑚. Then the functions 𝐸𝑎,𝜆𝛼,𝛽(𝑡),𝛽=𝛼𝑚+1,,𝛼1,𝛼(4.16) define eigenfunctions of the operator 𝑎𝛼(𝑞,) on each set 𝕋[𝜎(𝑎),𝑏]𝜎(𝑎)(𝑞,), where 𝕋𝑏𝜎(𝑎)(𝑞,) is satisfying |𝜆|(𝜈(𝑏))𝛼<1.

Proof. The assertion follows from Theorem 4.4 by the use of 𝜂=𝛼.

Our final aim is to show that any solution of (4.1) can be written as a linear combination of (𝑞,)-Mittag-Leffler functions (4.16).

Lemma 4.6. Let 𝛼+ and 𝑚+ be such that 𝑚1<𝛼𝑚. Then 𝑊𝑚𝛼𝐸𝑎,𝜆𝛼,𝛼𝑚+1,𝐸𝑎,𝜆𝛼,𝛼𝑚+2,,𝐸𝑎,𝜆𝛼,𝛼(𝜎𝑚(𝑎))=𝑚𝑘=11𝜈𝜎1𝜆𝑘(𝑎)𝛼0.(4.17)

Proof. The case 𝑚=1 is trivial. For 𝑚2, we can formally write 𝜆𝐸𝑎,𝜆𝛼,𝛼(𝑡)=𝐸𝑎,𝜆𝛼,(𝑡) for all 𝕋𝑡𝜎𝑚(𝑎)(𝑞,)(=0,,𝑚2). Consequently, applying Theorem 4.4, the Wronskian can be expressed as 𝑊𝑚𝛼𝐸𝑎,𝜆𝛼,𝛼𝑚+1,𝐸𝑎,𝜆𝛼,𝛼𝑚+2,,𝐸𝑎,𝜆𝛼,𝛼(𝜎𝑚(𝑎))=det𝑀𝑚(𝜎𝑚(𝑎)),(4.18) where 𝑀𝑚(𝜎𝑚𝐸(𝑎))=𝑎,𝜆𝛼,1(𝜎𝑚(𝑎))𝐸𝑎,𝜆𝛼,2(𝜎𝑚(𝑎))𝐸𝑎,𝜆𝛼,𝑚(𝜎𝑚𝐸(𝑎))𝑎,𝜆𝛼,0(𝜎𝑚(𝑎))𝐸𝑎,𝜆𝛼,1(𝜎𝑚(𝑎))𝐸𝑎,𝜆𝛼,𝑚1(𝜎𝑚𝐸(𝑎))𝑎,𝜆𝛼,2𝑚(𝜎𝑚(𝑎))𝐸𝑎,𝜆𝛼,3𝑚(𝜎𝑚(𝑎))𝐸𝑎,𝜆𝛼,1(𝜎𝑚(𝑎)).(4.19) Using the 𝑞-Pascal rule (2.15), we obtain the equality 𝐸𝑎,𝜆𝛼,𝑖(𝜎𝑚(𝑎))𝜈(𝜎(𝑎))𝐸𝑎,𝜆𝛼,𝑖1(𝜎𝑚(𝑎))=𝐸𝜎(𝑎),𝜆𝛼,𝑖(𝜎𝑚(𝑎)),𝑖,𝑖3𝑚.(4.20) Starting with the first row, (𝑚2) elementary row operations of the type (4.20) transform the matrix 𝑀𝑚(𝜎𝑚(𝑎)) into the matrix 𝑀𝑚(𝜎𝑚𝐸(𝑎))=𝜎𝑚1(𝑎),𝜆𝛼,1(𝜎𝑚(𝑎))𝐸𝜎𝑚1(𝑎),𝜆𝛼,2(𝜎𝑚(𝑎))𝐸𝜎𝑚1(𝑎),𝜆𝛼,𝑚(𝜎𝑚𝐸(𝑎))𝜎𝑚2(𝑎),𝜆𝛼,0(𝜎𝑚(𝑎))𝐸𝜎𝑚2(𝑎),𝜆𝛼,1(𝜎𝑚(𝑎))𝐸𝜎𝑚2(𝑎),𝜆𝛼,𝑚1(𝜎𝑚𝐸(𝑎))𝑎,𝜆𝛼,2𝑚(𝜎𝑚(𝑎))𝐸𝑎,𝜆𝛼,3𝑚(𝜎𝑚(𝑎))𝐸𝑎,𝜆𝛼,1(𝜎𝑚(𝑎))(4.21) with the property 𝑀det𝑚(𝜎𝑚(𝑎))=det𝑀𝑚(𝜎𝑚(𝑎)). By Lemma 2.3, we have 𝐸𝜎𝑖(𝑎),𝜆𝛼,𝑝(𝜎𝑚(𝑎))𝜈(𝜎𝑚(𝑎))𝐸𝜎𝑖(𝑎),𝜆𝛼,𝑝1(𝜎𝑚(𝑎))=𝐸𝜎𝑖(𝑎),𝜆𝛼,𝑝𝜎𝑚1𝐸(𝑎),𝑖=0,,𝑚2,𝜎𝑖(𝑎),𝜆𝛼,𝑝(𝜎𝑚(𝑎))𝜈(𝜎𝑚(𝑎))𝐸𝜎𝑖(𝑎),𝜆𝛼,𝑝1(𝜎𝑚(𝑎))=0,𝑖=𝑚1,(4.22) where 𝑝,𝑝3𝑚+𝑖. Starting with the last column, using 𝑚1 elementary column operations of the type (4.22), we obtain the matrix 𝐸𝜎𝑚1(𝑎),𝜆𝛼,1(𝜎𝑚𝐸(𝑎))𝜎𝑚2(𝑎),𝜆𝛼,0(𝜎𝑚𝐸(𝑎))𝑎,𝜆𝛼,2𝑚(𝜎𝑚(𝑎))00𝑀𝑚1𝜎𝑚1(𝑎)(4.23) preserving the value of 𝑀det𝑚(𝜎𝑚(𝑎)). Since 𝐸𝜎𝑚1(𝑎),𝜆𝛼,1(𝜎𝑚(𝑎))=𝑘=0𝜆𝑘(𝜈(𝜎𝑚(𝑎)))𝛼𝑘=11𝜆(𝜈(𝜎𝑚(𝑎)))𝛼,(4.24) we can observe the recurrence 𝑀det𝑚(𝜎𝑚1(𝑎))=1𝜆(𝜎𝑚(𝑎))𝛼𝑀det𝑚1𝜎𝑚1(𝑎),(4.25) which implies the assertion.

Now we summarize the results of Theorem 3.9, Corollary 4.5, and Lemma 4.6 to obtain

Theorem 4.7. Let 𝑦(𝑡) be any solution of (4.1) defined on 𝕋[𝜎(𝑎),𝑏]𝜎(𝑎)(𝑞,), where 𝕋𝑏𝜎(𝑎)(𝑞,) is satisfying |𝜆|(𝜈(𝑏))𝛼<1. Then 𝑦(𝑡)=𝑚𝑗=1𝑐𝑗𝐸𝑎,𝜆𝛼,𝛼𝑚+𝑗(𝑡),(4.26) where 𝑐1,,𝑐𝑚 are real constants.

We conclude this paper by the illustrating example.

Example 4.8. Consider the initial value problem 𝑎𝛼(𝑞,)𝑦(𝑡)=𝜆𝑦(𝑡),𝜎3(𝑎)𝑡𝜎𝑛(𝑎),1<𝛼2,𝑎𝛼1(𝑞,)||𝑦(𝑡)𝑡=𝜎2(𝑎)=𝑦𝛼1,𝑎𝛼2(𝑞,)||𝑦(𝑡)𝑡=𝜎2(𝑎)=𝑦𝛼2,(4.27) where 𝑛 is a positive integer given by the condition |𝜆|𝜈(𝜎𝑛(𝑎))𝛼<1. By Theorem 4.7, its solution can be expressed as a linear combination 𝑦(𝑡)=𝑐1𝐸𝑎,𝜆𝛼,𝛼1(𝑡)+𝑐2𝐸𝑎,𝜆𝛼,𝛼(𝑡).(4.28) The constants 𝑐1, 𝑐2 can be determined from the system 𝑉2𝛼𝐸𝑎,𝜆𝛼,𝛼1,𝐸𝑎,𝜆𝛼,𝛼𝜎2𝑐(𝑎)1𝑐2=𝑦𝛼2𝑦𝛼1(4.29) with the matrix elements 𝑣11=𝑣22=[1]𝑞+[𝛼]𝑞[1]𝑞𝜆𝜈(𝜎(𝑎))𝛼(1𝜆𝜈(𝜎(𝑎))𝛼)𝜎1𝜆𝜈2(𝑎)𝛼,𝑣12=[2]𝑞[𝛼]𝜈(𝜎(𝑎))+𝑞[2]𝑞𝜆𝜈(𝜎(𝑎))𝛼+1(1𝜆𝜈(𝜎(𝑎))𝛼)𝜎1𝜆𝜈2(𝑎)𝛼,𝑣21=[𝛼]𝑞𝜆𝜈(𝜎(𝑎))𝛼1(1𝜆𝜈(𝜎(𝑎))𝛼)𝜎1𝜆𝜈2(𝑎)𝛼.(4.30) By Lemma 4.6, the matrix 𝑉2𝛼(𝐸𝑎,𝜆𝛼,𝛼1,𝐸𝑎,𝜆𝛼,𝛼)(𝜎2(𝑎)) has a nonzero determinant, hence applying the Cramer rule, we get 𝑐1=𝑦𝛼2𝑣22𝑦𝛼1𝑣12𝑊2𝛼𝐸𝑎,𝜆𝛼,𝛼1,𝐸𝑎,𝜆𝛼,𝛼𝜎2,𝑐(𝑎)2=𝑦𝛼1𝑣11𝑦𝛼2𝑣21𝑊2𝛼𝐸𝑎,𝜆𝛼,𝛼1,𝐸𝑎,𝜆𝛼,𝛼𝜎2.(𝑎)(4.31)
Now we make a particular choice of the parameters 𝛼, 𝑎, 𝜆, 𝑦𝛼1 and 𝑦𝛼2 and consider the initial value problem in the form 11.8(𝑞,)1𝑦(𝑡)=3𝑦(𝑡),𝜎3(1)𝑡𝜎𝑛(1),10.8(𝑞,)|||𝑦(𝑡)𝑡=𝜎2(1)=1,10.2(𝑞,)|||𝑦(𝑡)𝑡=𝜎2(1)=1,(4.32) where 𝑛 is a positive integer satisfying 𝜈(𝜎𝑛(1))<35/9. If we take the time scale of integers (the case 𝑞==1), then the solution 𝑦(𝑡) of the corresponding initial value problem takes the form 𝑦(𝑡)=145𝑘=013𝑘𝑡2𝑗=1(𝑗+1.8𝑘0.2)2(𝑡2)!15𝑘=013𝑘𝑡2𝑗=1(𝑗+1.8𝑘+0.8)(𝑡2)!,𝑡=2,3,.(4.33) Similarly we can determine 𝑦(𝑡) for other choices of 𝑞 and . For comparative reasons, Figure 1 depicts (in addition to the above case 𝑞==1) the solution 𝑦(𝑡) under particular choices 𝑞=1.2, =0 (the pure 𝑞-calculus), 𝑞=1, =0.1 (the pure -calculus) and also the solution of the corresponding continuous (differential) initial value problem.

Acknowledgments

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.