Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations

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,ℎ)-calculus, which can be reduced to ordinary difference calculus and 𝑞-difference calculus via the choice 𝑞=ℎ=1 and ℎ=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,ℎ)-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,ℎ)-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 ℎ_𝑛∶𝕋²→ℝ, 𝑛∈ℕ₀ are defined recursively as follows: ℎ₀(𝑡,𝑠)=1∀𝑠,𝑡∈𝕋(2.1) and, given ℎ_𝑛 for 𝑛∈ℕ₀, we have ℎ_𝑛+1(𝑡,𝑠)=^𝑡𝑠ℎ_𝑛(𝜏,𝑠)∇𝜏∀𝑠,𝑡∈𝕋.(2.2) Now let 𝑓∶𝕋→ℝ be ∇-integrable on [𝑎,𝑏]∩𝕋, 𝑎,𝑏∈𝕋. We put _𝑎∇⁻¹𝑓(𝑡)=^𝑡𝑎𝑓(𝜏)∇𝜏∀𝑡∈𝕋,𝑎≤𝑡≤𝑏(2.3) and define recursively _𝑎∇^−𝑛𝑓(𝑡)=^𝑡𝑎𝑎∇^−𝑛+1𝑓(𝜏)∇𝜏(2.4) for 𝑛=2,3,…. Then we have the following.

Proposition 2.1 (Nabla Cauchy formula). Let 𝑛∈ℤ⁺, 𝑎,𝑏∈𝕋 and let 𝑓∶𝕋→ℝ be ∇-integrable on [𝑎,𝑏]∩𝕋. If 𝑡∈𝕋, 𝑎≤𝑡≤𝑏, then _𝑎∇^−𝑛𝑓(𝑡)=^𝑡𝑎ℎ_𝑛−1(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 𝑛≥2 and assume that (2.5) holds with 𝑛 replaced with 𝑛−1, that is, _𝑎∇^−𝑛+1𝑓(𝑡)=^𝑡𝑎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+1𝑓(𝑡). We show that the right-hand side of (2.5) is an antiderivative of ∫^𝑡𝑎ℎ_𝑛−2(𝑡,𝜌(𝜏))𝑓(𝜏)∇𝜏. Indeed, it holds ∇^𝑡𝑎ℎ_𝑛−1(𝑡,𝜌(𝜏))𝑓(𝜏)∇𝜏=^𝑡𝑎∇ℎ_𝑛−1(𝑡,𝜌(𝜏))𝑓(𝜏)∇𝜏=^𝑡𝑎ℎ_𝑛−2(𝑡,𝜌(𝜏))𝑓(𝜏)∇𝜏,(2.7) where we have employed the property ∇^𝑡𝑎𝑔(𝑡,𝜏)∇𝜏=^𝑡𝑎∇𝑔(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 _𝑎∇^−𝛼𝑓(𝑡) for positive reals 𝛼. However, it requires a reasonable and natural extension of a discrete system of monomials (ℎ_𝑛,𝑛∈ℕ₀) 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 ℎ₁(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 ℎ_𝑛(𝑡,𝑠)=(𝑡−𝑠)^𝑛𝑛!,(2.9) while for discrete time scales 𝕋=ℤ and 𝕋=𝑞^ℤ={𝑞^𝑘,𝑘∈ℤ}∪{0}, 𝑞>1, we have ℎ_𝑛(𝑡,𝑠)=∏^{𝑛−1𝑗=0}(𝑡−𝑠+𝑗)𝑛!,ℎ^𝑛(𝑡,𝑠)=^𝑛−1_𝑗=0𝑞^𝑗𝑡−𝑠∑^𝑗𝑟=0𝑞^𝑟,(2.10) respectively. In this connection, we recall a conventional notation used in ordinary difference calculus and 𝑞-calculus, namely,(𝑡−𝑠)^(𝑛)=^𝑛−1_𝑗=0(𝑡−𝑠+𝑗),(𝑡−𝑠)^(𝑛)̃𝑞=𝑡^𝑛𝑛−1_𝑗=01−𝑞̃^𝑗𝑠𝑡(0<𝑞̃<1)(2.11) and [𝑗]_𝑞=∑^{𝑗−1𝑟=0}𝑞^𝑟(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∈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<𝑞̃<1 as Γ_̃𝑞(𝑥)=(𝑞̃,𝑞̃)^∞(1−𝑞̃)^1−𝑥(𝑞̃^𝑥,𝑞̃)_∞,(2.12) where (𝑝,𝑞̃)_∞=∏^∞𝑗=0(1−𝑝𝑞̃^𝑗), 𝑥∈ℝ⧵{0,−1,−2,…}. Note that this function satisfies the functional relation Γ_̃𝑞(𝑥+1)=[𝑥]_̃𝑞Γ_̃𝑞(𝑥) and the condition Γ_̃𝑞(1)=1. Using this, the 𝑞-binomial coefficient can be introduced as 𝑥𝑘_̃𝑞=Γ^̃𝑞(𝑥+1)Γ_̃𝑞(𝑘+1)Γ_̃𝑞(𝑥−𝑘+1),𝑥∈ℝ,𝑘∈ℤ.(2.13) Note that although the 𝑞-Gamma function is not defined at nonpositive integers, the formula Γ_̃𝑞(𝑥+𝑚)Γ_̃𝑞(𝑥)=(−1)^𝑚𝑞̃^{𝑚2𝑥𝑚+}Γ_̃𝑞(1−𝑥)Γ_̃𝑞(1−𝑥−𝑚),𝑥∈ℝ,𝑚∈ℤ⁺(2.14) permits to calculate this ratio also at such the points. It is well known that if 𝑞̃→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𝑥𝑘_̃𝑞=𝑥𝑘−−11_̃𝑞+𝑞̃^𝑘𝑥−𝑘1_̃𝑞𝑥𝑘,𝑥∈ℝ,𝑘∈ℤ,(2.15)_̃𝑞=𝑞̃^𝑥−𝑘𝑥𝑘−−11_̃𝑞+𝑥−𝑘1_̃𝑞,𝑥∈ℝ,𝑘∈ℤ(2.16) and the 𝑞-Vandermonde identity ^𝑚_𝑗=0𝑚𝑥−𝑗_̃𝑞𝑦𝑗_̃𝑞𝑞̃^{𝑗2−𝑚𝑗+𝑥𝑗}=𝑥𝑚+𝑦_̃𝑞,𝑥,𝑦∈ℝ,𝑚∈ℕ₀(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 𝕋^𝑡0_(𝑞,ℎ)=𝑡₀𝑞^𝑘+[𝑘]_𝑞ℎ,𝑘∈ℤ∪ℎ1−𝑞,𝑡⁰>0,𝑞≥1,ℎ≥0,𝑞+ℎ>1(2.18) (see also [7]). Note that if 𝑞=1 then the cluster point ℎ/(1−𝑞)=−∞ is not involved in 𝕋^𝑡0_(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⁻¹𝜇(𝑡), respectively. In particular, if 𝑡₀=𝑞=ℎ=1, then 𝕋^𝑡0_(𝑞,ℎ) becomes ℤ, and if 𝑡₀=1, 𝑞>1, ℎ=0, then 𝕋^𝑡0_(𝑞,ℎ) is reduced to 𝑞^ℤ.

Let 𝑎∈𝕋^𝑡0_(𝑞,ℎ), 𝑎>ℎ/(1−𝑞) be fixed. Then we introduce restrictions of the time scale 𝕋^𝑡0_(𝑞,ℎ) by the relation 𝕋^{𝜎𝑖(𝑎)(𝑞,ℎ)}=𝑡∈𝕋^𝑡0_(𝑞,ℎ),𝑡≥𝜎^𝑖(𝑎),𝑖=0,1,…,(2.19) 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 𝑞̃=1/𝑞 whenever considering the time scale 𝕋^𝑡0_(𝑞,ℎ) or 𝕋^{𝜎𝑖(𝑎)(𝑞,ℎ)}.

Using the induction principle, we can verify that Taylor monomials on 𝕋^𝑡0_(𝑞,ℎ) have the form ℎ_𝑛(𝑡,𝑠)=∏^{𝑛−1𝑗=0}𝜎^𝑗(𝑡)−𝑠[𝑛]_𝑞!=∏^{𝑛−1𝑗=0}𝑡−𝜌^𝑗(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,ℎ)-power(𝑡−𝑠)^((𝑛)𝑞̃_,ℎ)=^𝑛−1_𝑗=0𝑡−𝜌^𝑗(𝑠)(2.21) unifying (2.11), then the Cauchy formula (2.5) can be rewritten for 𝕋=𝕋^𝑡0_(𝑞,ℎ) as _𝑎∇^−𝑛𝑓(𝑡)=^𝑡𝑎(𝑡−𝜌(𝜏))^{(𝑛−1)}𝑞̃_,ℎ[𝑛−1]_̃𝑞!𝑓(𝜏)∇𝜏.(2.22)

Discussing a reasonable generalization of (𝑞,ℎ)-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 (𝑡−𝑠)^(𝛼)=Γ(𝑡−𝑠+𝛼)Γ(𝑡−𝑠),(𝑡−𝑠)^(𝛼)̃𝑞=𝑡^𝛼(𝑠/𝑡,𝑞̃)_∞(𝑞̃^𝛼𝑠/𝑡,𝑞̃)_∞,𝑡≠0.(2.23) Now, we assume 𝑠,𝑡∈𝕋^𝑡0_(𝑞,ℎ), 𝑡≥𝑠>ℎ/(1−𝑞). First, consider (𝑞,ℎ)-power (2.21) corresponding to the time scale 𝕋^𝑡0_(𝑞,ℎ), where 𝑞>1. Then we can rewrite (2.21) as (𝑡−𝑠)^(𝑛)𝑞̃_,ℎ=𝑡+ℎ𝑞̃1−𝑞̃^𝑛𝑛−1_𝑗=01−𝑞̃^𝑗𝑠+ℎ𝑞̃/(1−𝑞̃)𝑡+ℎ𝑞̃/(1−𝑞̃)=([𝑡]−[𝑠])^(𝑛)̃𝑞,(2.24) where [𝑡]=𝑡+ℎ𝑞̃/(1−𝑞̃) and [𝑠]=𝑠+ℎ𝑞̃/(1−𝑞̃). A required extension of (𝑞,ℎ)-power (2.21) is then provided by the formula (𝑡−𝑠)^((𝛼)𝑞̃_,ℎ)=([𝑡]−[𝑠])^(𝛼)̃𝑞.(2.25) Now consider (𝑞,ℎ)-power (2.21) corresponding to the time scale 𝕋^𝑡0_(𝑞,ℎ), where 𝑞=1. Then (𝑡−𝑠)^{(𝑛)(1,ℎ)}=^𝑛−1_𝑗=0(𝑡−𝑠+𝑗ℎ)=ℎ^𝑛𝑛−1_𝑗=0𝑡−ℎ𝑠+𝑗=ℎ^𝑛((𝑡−𝑠)/ℎ+𝑛−1)!((𝑡−𝑠)/ℎ−1)!(2.26) and the formula (2.21) can be extended by (𝑡−𝑠)^{(𝛼)(1,ℎ)}=ℎ^𝛼Γ((𝑡−𝑠)/ℎ+𝛼)Γ((𝑡−𝑠)/ℎ).(2.27) These definitions are consistent, since it can be shown that 𝑞l̃im_→1−([𝑡]−[𝑠])^(𝛼)̃𝑞=(𝑡−𝑠)^{(𝛼)(1,ℎ)}.(2.28) Now the required extension of the monomial ℎ_𝑛(𝑡,𝑠) corresponding to 𝕋^𝑡0_(𝑞,ℎ) takes the form ℎ_𝛼(𝑡,𝑠)=(𝑡−𝑠)𝑞^((𝛼)̃_,ℎ)Γ_̃𝑞(𝛼+1).(2.29)

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

Proposition 2.2. Let 𝛼∈ℝ, 𝑠,𝑡∈𝕋^𝑡0_(𝑞,ℎ) and 𝑛∈ℕ₀ be such that 𝑡=𝜎^𝑛(𝑠). Then ℎ_𝛼(𝑡,𝑠)=(𝜈(𝑡))^𝛼𝛼+𝑛−𝑛−11_̃𝑞=(𝜈(𝑡))^𝛼−𝑛𝛼−−11_̃𝑞(−1)^𝑛−1𝑞̃^{𝑛2𝛼(𝑛−1)+}.(2.30)

Proof. Let 𝑞>1. Using the relations [𝑡]=𝜈(𝑡)(1−𝑞̃),[[𝑠𝑡]]=𝑞̃^𝑛,(2.31) we can derive that ℎ_𝛼(𝑡,𝑠)=[𝑡]^𝛼([𝑠]/[𝑡],𝑞̃)_∞Γ_̃𝑞(𝛼+1)(𝑞̃^𝛼[𝑠]/[𝑡],𝑞̃)_∞=(1−𝑞̃)^−𝛼𝜈(𝑡)^𝛼(𝑞̃^𝑛,𝑞̃)_∞Γ_̃𝑞(𝛼+1)𝑞̃^𝛼+𝑛,𝑞̃_∞=(𝜈(𝑡))^𝛼Γ_̃𝑞(𝛼+𝑛)Γ_̃𝑞(𝛼+1)Γ_̃𝑞(𝑛)=(𝜈(𝑡))^𝛼𝛼+𝑛−𝑛−11_̃𝑞.(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^𝑡0_(𝑞,ℎ), defined for 𝑚=1 as ∇_(𝑞,ℎ)𝑓(𝑡)=𝑓(𝑡)−𝑓(𝜌(𝑡))𝜈(𝑡)=𝑓((𝑡)1−−𝑓𝑞̃)(𝑡𝑞̃+(𝑡𝑞−̃ℎℎ))(2.33) and iteratively for higher orders.

Lemma 2.3. Let 𝑚∈ℤ⁺, 𝛼∈ℝ, 𝑠,𝑡∈𝕋^𝑡0_(𝑞,ℎ) and 𝑛∈ℤ⁺, 𝑛≥𝑚 be such that 𝑡=𝜎^𝑛(𝑠). Then ∇^{𝑚(𝑞,ℎ)}ℎ_𝛼(𝑡,𝑠)=ℎ^𝛼−𝑚0,(𝑡,𝑠),𝛼𝛼∈∉{{00,,11,,……,,𝑚𝑚−−11}}.,(2.34)

Proof. First let 𝑚=1. For 𝛼=0 we get ℎ₀(𝑡,𝑠)=1 and the first nabla (𝑞,ℎ)-derivative is zero. If 𝛼≠0, then by (2.30) and (2.16), we have ∇_(𝑞,ℎ)ℎ_𝛼(𝑡,𝑠)=ℎ^𝛼(𝑡,𝑠)−ℎ_𝛼(𝜌(𝑡),𝑠)=1𝜈(𝑡)𝜈(𝑡)(𝜈(𝑡))^𝛼𝛼+𝑛−𝑛−11_̃𝑞−(𝜈(𝜌(𝑡)))^𝛼=(𝜈(𝑡))_𝛼−1𝛼+𝑛−𝑛−22_̃𝑞𝛼+𝑛−𝑛−11_̃𝑞−𝑞̃^𝛼𝛼+𝑛−𝑛−22_̃𝑞=ℎ^𝛼−1(𝑡,𝑠).(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 _𝑎∇^{−𝛼(𝑞,ℎ)}𝑓(𝑡)=^𝑡𝑎ℎ_𝛼−1(𝑡,𝜌(𝜏))𝑓(𝜏)∇𝜏(2.36) (see also [7]). The nabla (𝑞,ℎ)-fractional derivative of order 𝛼∈ℝ⁺ is then defined by _𝑎∇^{𝛼(𝑞,ℎ)}𝑓(𝑡)=∇^{𝑚𝑎(𝑞,ℎ)}∇^{−(𝑚−𝛼)(𝑞,ℎ)}𝑓(𝑡),(2.37) 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.

3. A Linear Initial Value Problem

In this section, we are going to discuss the linear initial value problem ^𝑚_𝑗=1𝑝_{𝑚−𝑗+1}(𝑡)_𝑎∇^{𝛼−𝑗+1(𝑞,ℎ)}𝑦(𝑡)+𝑝₀(𝑡)𝑦(𝑡)=0,𝑡∈𝕋^{𝜎𝑚+1(𝑎)(𝑞,ℎ)},(3.1)_𝑎∇^{𝛼−𝑗(𝑞,ℎ)}𝑦(𝑡)|||_{𝑡=𝜎𝑚(𝑎)}=𝑦_𝛼−𝑗,𝑗=1,2,…,𝑚,(3.2) where 𝛼∈ℝ⁺ and 𝑚∈ℤ⁺ are such that 𝑚−1<𝛼≤𝑚. Further, we assume that 𝑝_𝑗(𝑡) are arbitrary real-valued functions on 𝕋^{𝜎𝑚+1(𝑎)(𝑞,ℎ)}(𝑗=1,…,𝑚−1), 𝑝_𝑚(𝑡)=1 on 𝕋^{𝜎𝑚+1(𝑎)(𝑞,ℎ)} 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−1_𝑖=0𝑎_𝑖(𝑡)𝑦𝜌^𝑖(𝑡)=0,𝑡∈𝕋^{𝜎𝑚+1(𝑎)(𝑞,ℎ)},𝑛beingsuchthat𝑡=𝜎^𝑛(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),…,𝜎^𝑚(𝑎). In particular, if 𝑎₀(𝑡)≠0 for all 𝑡∈𝕋^{𝜎𝑚+1(𝑎)(𝑞,ℎ)}, then arbitrary values of 𝑦(𝜎(𝑎)),𝑦(𝜎²(𝑎)),…,𝑦(𝜎^𝑚(𝑎)) determine uniquely the solution 𝑦(𝑡) for all 𝑡∈𝕋^{𝜎𝑚+1(𝑎)(𝑞,ℎ)}. We show that the values 𝑦_𝛼−1,𝑦_𝛼−2,…,𝑦_𝛼−𝑚, introduced by (3.2), keep the same properties.

Proposition 3.1. Let 𝑦∶𝕋^{𝜎(𝑎)(𝑞,ℎ)}→ℝ be a function. Then (3.2) represents a one-to-one mapping between the vectors (𝑦(𝜎(𝑎)),𝑦(𝜎²(𝑎)),…,𝑦(𝜎^𝑚(𝑎))) and (𝑦_𝛼−1,𝑦_𝛼−2,…,𝑦_𝛼−𝑚).

Proof. The case 𝛼∈ℤ⁺ is well known from the literature. Let 𝛼∉ℤ⁺. We wish to show that the values of 𝑦(𝜎(𝑎)),𝑦(𝜎²(𝑎)),…,𝑦(𝜎^𝑚(𝑎)) determine uniquely the values of _𝑎∇^{𝛼−1(𝑞,ℎ)}𝑦(𝑡)||_{𝑡=𝜎𝑚(𝑎)},_𝑎∇^{𝛼−2(𝑞,ℎ)}𝑦(𝑡)||_{𝑡=𝜎𝑚(𝑎)},…,_𝑎∇^{𝛼−𝑚(𝑞,ℎ)}𝑦(𝑡)||_{𝑡=𝜎𝑚(𝑎)}(3.4) and vice versa. Utilizing the relation _𝑎∇^{𝛼−𝑗(𝑞,ℎ)}𝑦(𝑡)|||_{𝑡=𝜎𝑚(𝑎)}=^𝑚_𝑘=1𝜈𝜎^{𝑚−𝑘+1}(𝑎)ℎ_{𝑗−1−𝛼}𝜎^𝑚(𝑎),𝜎^𝑚−𝑘(𝑎)𝑦𝜎^{𝑚−𝑘+1}(𝑎)(3.5) (see [7, Propositions  1 and  3] with respect to (2.30)), we can rewrite (3.2) as the linear mapping ^𝑚_𝑘=1𝑟_𝑗𝑘𝑦𝜎^{𝑚−𝑘+1}(𝑎)=𝑦_𝛼−𝑗,𝑗=1,…,𝑚,(3.6) where 𝑟_𝑗𝑘=𝜈𝜎^{𝑚−𝑘+1}(𝑎)ℎ_{𝑗−1−𝛼}𝜎^𝑚(𝑎),𝜎^𝑚−𝑘(𝑎),𝑗,𝑘=1,…,𝑚(3.7) are elements of the transformation matrix 𝑅_𝑚. We show that 𝑅_𝑚 is regular. Obviously, det𝑅_𝑚=^𝑚_𝑘=1𝜈𝜎^𝑘(𝑎)det𝐻_𝑚,(3.8) where 𝐻_𝑚=⎛⎜⎜⎜⎜⎝ℎ^−𝛼ℎ𝜎^𝑚(𝑎),𝜎^𝑚−1(𝑎)ℎ_−𝛼𝜎^𝑚(𝑎),𝜎^𝑚−2(𝑎)⋯ℎ_−𝛼(𝜎^𝑚(𝑎),𝑎)_1−𝛼ℎ𝜎^𝑚(𝑎⋮),𝜎^𝑚−1(𝑎)ℎ_1−𝛼𝜎^𝑚(𝑎⋮),𝜎^𝑚−2(𝑎)⋱⋯ℎ_1−𝛼(𝜎⋮^𝑚_{𝑚−1−𝛼}(𝑎),𝑎)𝜎^𝑚(𝑎),𝜎^𝑚−1(𝑎)ℎ_{𝑚−1−𝛼}𝜎^𝑚(𝑎),𝜎^𝑚−2(𝑎)⋯ℎ_{𝑚−1−𝛼}(𝜎^𝑚(𝑎),𝑎)⎞⎟⎟⎟⎟⎠.(3.9) To calculate det𝐻_𝑚, we employ some elementary operations preserving the value of det𝐻_𝑚. Using the properties ℎ_𝑖−𝛼𝜎^𝑚ℎ(𝑎),𝜎^ℓ(𝑎)−𝜈(𝜎^𝑚(𝑎))ℎ(𝑖_{𝑖−𝛼−1}=𝜎1^𝑚,(2𝑎,)…,𝜎,^ℓ𝑚(𝑎−)1=,𝑙ℎ=_𝑖−𝛼0,𝜎1,…^𝑚−1𝑚(𝑎)−,𝜎2)^ℓ,_𝑖−𝛼(𝑎)𝜎^𝑚(𝑎),𝜎^𝑚−1(𝑎)−𝜈(𝜎^𝑚(𝑎))ℎ_{𝑖−𝛼−1}𝜎^𝑚(𝑎),𝜎^𝑚−1(𝑎)=0,(3.10) which follow from Lemma 2.3, we multiply the 𝑖th row (𝑖=1,2,…,𝑚−1) of 𝐻_𝑚 by −𝜈(𝜎^𝑚(𝑎)) and add it to the successive one. We arrive at the form ⎛⎜⎜⎜⎜⎝ℎ_−𝛼𝜎^𝑚(𝑎⋮00),𝜎^𝑚−1(𝑎)ℎ_−𝛼𝜎^𝑚(𝑎),𝜎^𝑚−2(𝑎)⋯ℎ_−𝛼(𝜎^𝑚(𝑎),𝑎)𝐻_𝑚−1⎞⎟⎟⎟⎟⎠.(3.11) Then we apply repeatedly this procedure to obtain the triangular matrix ⎛⎜⎜⎜⎜⎝ℎ_−𝛼𝜎^𝑚(𝑎0),𝜎^𝑚−1(𝑎)ℎℎ_−𝛼𝜎^𝑚(𝑎),𝜎^𝑚−2(𝑎)⋯ℎ_−𝛼(𝜎^𝑚(𝑎),𝑎)_1−𝛼⋮0𝜎^𝑚−1(⋮0𝑎),𝜎^𝑚−2(𝑎)⋱⋯⋯ℎℎ_1−𝛼𝜎_{𝑚−1−𝛼}⋮^𝑚−1(𝑎),𝑎(𝜎(𝑎),𝑎)⎞⎟⎟⎟⎟⎠.(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 det𝐻_𝑚=^𝑚_𝑘=1𝜈𝜎^𝑘(𝑎)^{𝑚−𝑘−𝛼},thatis,det𝑅_𝑚=^𝑚_𝑘=1𝜈𝜎^𝑘(𝑎)^{𝑚−𝑘−𝛼+1}≠0.(3.13) 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]).

Definition 3.2. An 𝑛×𝑛-matrix-valued function 𝐴(𝑡) on a time scale 𝕋 is called 𝜈-regressive provided det(𝐼−𝜈(𝑡)𝐴(𝑡))≠0∀𝑡∈𝕋_𝜅,(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∈ℝ⁺ and 𝑚∈ℤ⁺ be such that 𝑚−1<𝛼≤𝑚. Then (3.1) is called 𝜈-regressive provided the matrix 𝐴(𝑡)=⎛⎜⎜⎜⎜⎜⎜⎝−𝑝⋮000⋮100⋱⋯01⋱⋱⋯0⋮001₀(𝑡)𝜈^𝑚−𝛼(𝑡)−𝑝¹(𝑡)⋯−𝑝_𝑚−2(𝑡)−𝑝_𝑚−1(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+^𝑚−1_𝑗=1𝑝_𝑚−𝑗(𝑡)(𝜈(𝑡))^𝑗+𝑝₀(𝑡)(𝜈(𝑡))^𝛼≠0∀𝑡∈𝕋^{𝜎𝑚+1(𝑎)(𝑞,ℎ)}.(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 𝑡∈𝕋^{𝜎(𝑎)(𝑞,ℎ)}.

Proof. The conditions (3.2) enable us to determine the values of 𝑦(𝜎(𝑎)),𝑦(𝜎²(𝑎)),…,𝑦(𝜎^𝑚(𝑎)) by the use of (3.6). To calculate the values of 𝑦(𝜎^𝑚+1(𝑎)),𝑦(𝜎^𝑚+2(𝑎)),…, we perform the transformation 𝑧_𝑗(𝑡)=_𝑎∇^{𝛼−𝑚+𝑗−1(𝑞,ℎ)}𝑦(𝑡),𝑡∈𝕋^{𝜎𝑗(𝑎)(𝑞,ℎ)},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 _𝑎∇^{𝑎𝑚−𝛼(𝑞,ℎ)}∇^{−(𝑚−𝛼)(𝑞,ℎ)}𝑦(𝑡)=𝑦(𝑡) (see [7]) and expanding the fractional derivative, we arrive at 𝑦(𝑡)=_𝑎∇^{𝑚−𝛼(𝑞,ℎ)}𝑧₁(𝑡)=𝑧¹(𝑡)𝜈^𝑚−𝛼(𝑡)+^{𝑎𝜌(𝑡)}ℎ_{𝛼−𝑚−1}(𝑡,𝜌(𝜏))𝑧₁(𝜏)∇𝜏.(3.19) Therefore, the problem (3.1)-(3.2) can be rewritten to the vector form _𝑎∇_(𝑞,ℎ)𝑧𝑧((𝑡𝜎)_𝑚=𝐴(𝑡)𝑧(𝑡)+𝑏(𝑡),𝑡∈𝕋^{𝜎𝑚+1(𝑎)(𝑞,ℎ)},(𝑎))=𝑦_𝛼−𝑚,…,𝑦_𝛼−1^𝑇,(3.20) where 𝑧(𝑡)=𝑧₁(𝑡),…,𝑧_𝑚(𝑡)^𝑇,𝑏(𝑡)=0,…,0,−𝑝₀(𝑡)^{𝑎𝜌(𝑡)}ℎ_{𝛼−𝑚−1}(𝑡,𝜌(𝜏))𝑧₁(𝜏)∇𝜏^𝑇(3.21) and 𝐴(𝑡) is given by (3.16). The 𝜈-regressivity of the matrix 𝐴(𝑡) enables us to write 𝑧(𝑡)=(𝐼−𝜈(𝑡)𝐴(𝑡))⁻¹(𝑧(𝜌(𝑡))+𝜈(𝑡)𝑏(𝑡)),𝑡∈𝕋^{𝜎𝑚+1(𝑎)(𝑞,ℎ)},(3.22) 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(𝑦₁,…,𝑦_𝑚)(𝑡) as determinant of the matrix 𝑉_𝛾𝑦₁,…,𝑦_𝑚(𝑡)=⎛⎜⎜⎜⎜⎝^𝑎∇^{−𝛾(𝑞,ℎ)}𝑦₁(𝑡)_𝑎∇^{−𝛾(𝑞,ℎ)}𝑦₂(𝑡)⋯_𝑎∇^{−𝛾(𝑞,ℎ)}𝑦_𝑚(𝑡)_𝑎∇⋮^{1−𝛾(𝑞,ℎ)}𝑦₁(𝑡)_𝑎∇⋮^{1−𝛾(𝑞,ℎ)}𝑦₂(𝑡)⋱⋯_𝑎∇⋮^{1−𝛾(𝑞,ℎ)}_𝑎𝑦_𝑚(𝑡)∇^{𝑚−1−𝛾(𝑞,ℎ)}𝑦₁(𝑡)_𝑎∇^{𝑚−1−𝛾(𝑞,ℎ)}𝑦₂(𝑡)⋯_𝑎∇^{𝑚−1−𝛾(𝑞,ℎ)}𝑦_𝑚(𝑡)⎞⎟⎟⎟⎟⎠,𝑡∈𝕋^{𝜎𝑚(𝑎)(𝑞,ℎ)}.(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 𝑊_𝛾(𝑦₁,…,𝑦_𝑚)(𝑡) coincides for 𝛾=0 with the classical definition of the Wronskian (see [8]). Moreover, it holds 𝑊_𝛾(𝑦₁,…,𝑦_𝑚)(𝑡)=𝑊₀(_𝑎∇^{−𝛾(𝑞,ℎ)}𝑦₁,…,_𝑎∇^{−𝛾(𝑞,ℎ)}𝑦_𝑚)(𝑡).

Theorem 3.9. Let functions 𝑦₁(𝑡),…,𝑦_𝑚(𝑡) be solutions of the 𝜈-regressive equation (3.1) and let 𝑊_𝑚−𝛼(𝑦₁,…,𝑦_𝑚)(𝜎^𝑚(𝑎))≠0. Then any solution 𝑦(𝑡) of (3.1) can be written in the form 𝑦(𝑡)=^𝑚_𝑘=1𝑐_𝑘𝑦_𝑘(𝑡),𝑡∈𝕋^{𝜎(𝑎)(𝑞,ℎ)},(3.24) where 𝑐₁,…,𝑐_𝑚 are real constants.

Proof. Let 𝑦(𝑡) be a solution of (3.1). By Proposition 3.1, there exist real scalars 𝑦_𝛼−1,…,𝑦_𝛼−𝑚 such that 𝑦(𝑡) is satisfying (3.2). Now we consider the function 𝑢(𝑡)=∑^𝑚𝑘=1𝑐_𝑘𝑦_𝑘(𝑡), where the 𝑚-tuple (𝑐₁,…,𝑐_𝑚) is the unique solution of 𝑉_𝑚−𝛼𝑦₁,…,𝑦_𝑚(𝜎^𝑚(𝑎))⋅⎛⎜⎜⎜⎝𝑐¹𝑐₂𝑐⋮_𝑚⎞⎟⎟⎟⎠=⎛⎜⎜⎜⎝𝑦^𝛼−𝑚𝑦_{𝛼−𝑚+1}𝑦⋮_𝛼−1⎞⎟⎟⎟⎠.(3.25) The linearity of (3.1) implies that 𝑢(𝑡) has to be its solution. Moreover, it holds _𝑎∇^{𝛼−𝑗(𝑞,ℎ)}𝑢(𝑡)|||_{𝑡=𝜎𝑚(𝑎)}=𝑦_𝛼−𝑗,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 𝑡∈𝕋^{𝜎(𝑎)(𝑞,ℎ)} 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,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 _𝑎∇^{𝛼(𝑞,ℎ)}𝑦(𝑡)=𝜆𝑦(𝑡),𝜆∈ℝ,𝑡∈𝕋^{𝜎𝑚+1(𝑎)(𝑞,ℎ)}.(4.1) Throughout this section, we assume that 𝜈-regressivity condition for (4.1) is ensured, that is, 𝜆(𝜈(𝑡))^𝛼≠1.(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 [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.

Proposition 4.1. Let 𝛼∈ℝ⁺, 𝛽∈ℝ and 𝑡∈𝕋^{𝜎(𝑎)(𝑞,ℎ)}. Then it holds _𝑎∇^{−𝛼(𝑞,ℎ)}ℎ_𝛽(𝑡,𝑎)=ℎ_𝛼+𝛽(𝑡,𝑎).(4.3)

Proof. Let 𝑡∈𝕋^{𝜎(𝑎)(𝑞,ℎ)} be such that 𝑡=𝜎^𝑛(𝑎) for some 𝑛∈ℤ⁺. We have _𝑎∇^{−𝛼(𝑞,ℎ)}ℎ_𝛽(𝑡,𝑎)=^𝑛−1_𝑘=0=ℎ_𝛼−1𝑡,𝜌^𝑘+1(𝑡)𝜈𝜌^𝑘(𝑡)ℎ_𝛽𝜌^𝑘(𝑡),𝑎_𝑛−1_𝑘=0(𝜈(𝑡))^𝛼−1−𝑘𝛼_̃𝑞×𝜈𝜌(−1)^𝑘𝑞̃^{2(𝛼−1)𝑘+𝑘+1}𝑞̃^𝑘𝜈(𝑡)_𝑘(𝑡)^𝛽𝑛−−𝛽𝑘−−11_̃𝑞(−1)^{𝑛−𝑘−1}𝑞̃^{2𝛽(𝑛−𝑘−1)+𝑛−𝑘}=(𝜈(𝑡))^{𝛼+𝛽𝑛−1}_𝑘=0−𝑘𝛼_̃𝑞𝑛−−𝛽𝑘−−11_̃𝑞(−1)^𝑛−1𝑞̃^{𝑘2𝑛2−𝑘(𝑛−1)+𝑘𝛼++𝛽(𝑛−1)}=(𝜈(𝑡))^{𝛼+𝛽𝑛−1}_𝑘=0𝑛−−𝑘𝛼−1_̃𝑞−𝛽𝑘−1_̃𝑞×(−1)^𝑛−1𝑞̃^{(𝑛−𝑘−1)2𝑛2−(𝑛−𝑘−1)(𝑛−1)+(𝑛−𝑘−1)𝛼++𝛽(𝑛−1)}=(𝜈(𝑡))^{𝛼+𝛽𝑛−1}_𝑘=0𝑛−−𝑘𝛼−1_̃𝑞−𝛽𝑘−1_̃𝑞(−1)^𝑛−1𝑞̃^{𝑘2𝑛2−𝑘(𝑛−1)−𝑘𝛼+(𝛼+𝛽)(𝑛−1)+}=(𝜈(𝑡))^𝛼+𝛽−𝛼𝑛−−𝛽1−1_̃𝑞(−1)^𝑛−1𝑞̃^{𝑛2(𝛼+𝛽)(𝑛−1)+}=ℎ_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∈ℝ⁺, 𝛽∈ℝ, 𝑡∈𝕋^{𝜎𝑚+1(𝑎)(𝑞,ℎ)}, where 𝑚∈ℤ⁺ is satisfying 𝑚−1<𝛼≤𝑚. Then _𝑎∇^{𝛼(𝑞,ℎ)}ℎ_𝛽(𝑡,𝑎)=ℎ^𝛽−𝛼0,(𝑡,𝑎),𝛽𝛽−−𝛼𝛼∉∈{{−−11,,……,,−−𝑚𝑚}},.(4.5)

Proof. Proposition 4.1 implies that _𝑎∇^{𝛼(𝑞,ℎ)}ℎ_𝛽(𝑡,𝑎)=∇^{𝑚(𝑞,ℎ)}^𝑎∇^{−(𝑚−𝛼)(𝑞,ℎ)}ℎ_𝛽(𝑡,𝑎)=∇^{𝑚(𝑞,ℎ)}ℎ_{𝑚+𝛽−𝛼}(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,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 𝐸_𝛼,𝛽(𝑡)=^∞_𝑘=0𝑡^𝑘Γ(𝛼𝑘+𝛽),𝛼,𝛽∈ℝ⁺,𝑡∈ℝ.(4.7) The fractional calculus frequently employs (4.7), because the function 𝑡^𝛽−1𝐸_𝛼,𝛽(𝜆𝑡^𝛼)=^∞_𝑘=0𝜆^𝑘𝑡^{𝛼𝑘+𝛽−1}Γ(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=ℎ=1.

Definition 4.3. Let 𝛼,𝛽,𝜆∈ℝ. We introduce the (𝑞,ℎ)-Mittag-Leffler function 𝐸^{𝑠,𝜆𝛼,𝛽}(𝑡) by the series expansion 𝐸^{𝑠,𝜆𝛼,𝛽}(𝑡)=^∞_𝑘=0𝜆^𝑘ℎ_{𝛼𝑘+𝛽−1}(𝑡,𝑠)⎛⎜⎜⎝=^∞_𝑘=0𝜆^𝑘(𝑡−𝑠)^{((𝛼𝑘+𝛽−1)}𝑞̃_,ℎ)Γ_̃𝑞(𝛼𝑘+𝛽)⎞⎟⎟⎠,𝑠,𝑡∈𝕋^{𝑎(𝑞,ℎ)},𝑡≥𝑠.(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,ℎ)-Mittag-Leffler function 𝐸^𝑎,𝜆1,1(𝑡)=^𝑛−1_𝑘=011−𝜆𝜈𝜌^𝑘(𝑡),(4.10) where 𝑛∈ℤ⁺ satisfies 𝑡=𝜎^𝑛(𝑎), is a solution of the equation ∇_(𝑞,ℎ)𝑦(𝑡)=𝜆𝑦(𝑡),𝑡∈𝕋^{𝜎(𝑎)(𝑞,ℎ)},(4.11) 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.

Theorem 4.4. (i) Let 𝜂∈ℝ⁺ and 𝑡∈𝕋^{𝜎(𝑎)(𝑞,ℎ)}. Then _𝑎∇^{−𝜂(𝑞,ℎ)}𝐸^{𝑎,𝜆𝛼,𝛽}(𝑡)=𝐸^{𝑎,𝜆𝛼,𝛽+𝜂}(𝑡).(4.12)
(ii) Let 𝜂∈ℝ⁺, 𝑚∈ℤ⁺ be such that 𝑚−1<𝜂≤𝑚 and let 𝛼𝑘+𝛽−1∉{0,−1,…,−𝑚+1} for all 𝑘∈ℤ⁺. If 𝑡∈𝕋^{𝜎𝑚+1(𝑎)(𝑞,ℎ)}, then _𝑎∇^{𝜂(𝑞,ℎ)}𝐸^{𝑎,𝜆𝛼,𝛽}(𝑡)=𝐸^{𝑎,𝜆𝛼,𝛽−𝜂}𝜆𝐸(𝑡)_{𝑎,𝜆𝛼,𝛽−𝜂+𝛼},𝛽−𝜂∉{0,−1,…,−𝑚+1},(𝑡),𝛽−𝜂∈{0,−1,…,−𝑚+1}.(4.13)

Proof. The part (i) follows immediately from Proposition 4.1. Considering the part (ii), we can write _𝑎∇^{𝜂(𝑞,ℎ)}𝐸^{𝑎,𝜆𝛼,𝛽}(𝑡)=_𝑎∇^{𝜂∞(𝑞,ℎ)}_𝑘=0𝜆^𝑘ℎ_{𝛼𝑘+𝛽−1}(𝑡,𝑎)=^∞_𝑘=0𝜆^𝑘𝑎∇^{𝜂(𝑞,ℎ)}ℎ_{𝛼𝑘+𝛽−1}(𝑡,𝑎)(4.14) due to the absolute convergence property.
If 𝑘∈ℤ⁺, then Corollary 4.2 implies the relation _𝑎∇^{𝜂(𝑞,ℎ)}ℎ_{𝛼𝑘+𝛽−1}(𝑡,𝑎)=ℎ_{𝛼𝑘+𝛽−𝜂−1}(𝑡,𝑎)(4.15) due to the assumption 𝛼𝑘+𝛽−1∉{0,−1,…,−𝑚+1}. If 𝑘=0, then two possibilities may occur. If 𝛽−𝜂∉{0,−1,…,−𝑚+1}, we get (4.15) with 𝑘=0 which implies the validity of (4.13)1. If 𝛽−𝜂∈{0,−1,…,−𝑚+1}, the nabla (𝑞,ℎ)-fractional derivative of this term is zero and by shifting the index 𝑘, we obtain (4.13)2.

Corollary 4.5. Let 𝛼∈ℝ⁺ and 𝑚∈ℤ⁺ be such that 𝑚−1<𝛼≤𝑚. Then the functions 𝐸^{𝑎,𝜆𝛼,𝛽}(𝑡),𝛽=𝛼−𝑚+1,…,𝛼−1,𝛼(4.16) define eigenfunctions of the operator _𝑎∇^{𝛼(𝑞,ℎ)} on each set [𝜎(𝑎),𝑏]∩𝕋^{𝜎(𝑎)(𝑞,ℎ)}, where 𝑏∈𝕋^{𝜎(𝑎)(𝑞,ℎ)} 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,ℎ)-Mittag-Leffler functions (4.16).

Lemma 4.6. Let 𝛼∈ℝ⁺ and 𝑚∈ℤ⁺ be such that 𝑚−1<𝛼≤𝑚. Then 𝑊_𝑚−𝛼𝐸^{𝑎,𝜆𝛼,𝛼−𝑚+1},𝐸^{𝑎,𝜆𝛼,𝛼−𝑚+2},…,𝐸^{𝑎,𝜆𝛼,𝛼}(𝜎^𝑚(𝑎))=^𝑚_𝑘=111−𝜆𝜈𝜎^𝑘(𝑎)^𝛼≠0.(4.17)

Proof. The case 𝑚=1 is trivial. For 𝑚≥2, we can formally write 𝜆𝐸^{𝑎,𝜆𝛼,𝛼−ℓ}(𝑡)=𝐸^{𝑎,𝜆𝛼,−ℓ}(𝑡) for all 𝑡∈𝕋^{𝜎𝑚(𝑎)(𝑞,ℎ)}(ℓ=0,…,𝑚−2). Consequently, applying Theorem 4.4, the Wronskian can be expressed as 𝑊_𝑚−𝛼𝐸^{𝑎,𝜆𝛼,𝛼−𝑚+1},𝐸^{𝑎,𝜆𝛼,𝛼−𝑚+2},…,𝐸^{𝑎,𝜆𝛼,𝛼}(𝜎^𝑚(𝑎))=det𝑀_𝑚(𝜎^𝑚(𝑎)),(4.18) where 𝑀_𝑚(𝜎^𝑚(𝑎))=⎛⎜⎜⎜⎜⎝𝐸^{𝑎,𝜆𝛼,1}𝐸(𝜎^𝑚(𝑎))𝐸^{𝑎,𝜆𝛼,2}(𝜎^𝑚(𝑎))…𝐸^{𝑎,𝜆𝛼,𝑚}(𝜎^𝑚(𝑎))_{𝑎,𝜆𝛼,0}𝐸(…𝜎^𝑚(𝑎))𝐸^{𝑎,𝜆𝛼,1}(…𝜎^𝑚(𝑎))⋱…𝐸^{𝑎,𝜆𝛼,𝑚−1}…_{𝑎,𝜆𝛼,2−𝑚}(𝜎^𝑚(𝑎))(𝜎^𝑚(𝑎))𝐸^{𝑎,𝜆𝛼,3−𝑚}(𝜎^𝑚(𝑎))…𝐸^{𝑎,𝜆𝛼,1}(𝜎^𝑚(𝑎))⎞⎟⎟⎟⎟⎠.(4.19) Using the 𝑞-Pascal rule (2.15), we obtain the equality 𝐸^{𝑎,𝜆𝛼,𝑖}(𝜎^𝑚(𝑎))−𝜈(𝜎(𝑎))𝐸^{𝑎,𝜆𝛼,𝑖−1}(𝜎^𝑚(𝑎))=𝐸^{𝜎(𝑎),𝜆𝛼,𝑖}(𝜎^𝑚(𝑎)),𝑖∈ℤ,𝑖≥3−𝑚.(4.20) Starting with the first row, (^𝑚2) elementary row operations of the type (4.20) transform the matrix 𝑀_𝑚(𝜎^𝑚(𝑎)) into the matrix 𝑀_𝑚(𝜎^𝑚(𝑎))=⎛⎜⎜⎜⎜⎜⎝𝐸^{𝜎𝑚−1(𝑎),𝜆𝛼,1}𝐸(𝜎^𝑚(𝑎))𝐸^{𝜎𝑚−1(𝑎),𝜆𝛼,2}(𝜎^𝑚(𝑎))…𝐸^{𝜎𝑚−1(𝑎),𝜆𝛼,𝑚}(𝜎^𝑚(𝑎))_{𝜎𝑚−2(𝑎),𝜆𝛼,0}𝐸…(𝜎^𝑚(𝑎))𝐸^{𝜎𝑚−2(𝑎),𝜆𝛼,1}…(𝜎^𝑚(𝑎))⋱…𝐸^{𝜎𝑚−2(𝑎),𝜆𝛼,𝑚−1}…_{𝑎,𝜆𝛼,2−𝑚}(𝜎^𝑚(𝑎))(𝜎^𝑚(𝑎))𝐸^{𝑎,𝜆𝛼,3−𝑚}(𝜎^𝑚(𝑎))…𝐸^{𝑎,𝜆𝛼,1}(𝜎^𝑚(𝑎))⎞⎟⎟⎟⎟⎟⎠(4.21) with the property det𝑀_𝑚(𝜎^𝑚(𝑎))=det𝑀_𝑚(𝜎^𝑚(𝑎)). By Lemma 2.3, we have 𝐸^{𝜎𝑖(𝑎),𝜆𝛼,𝑝}(𝜎^𝑚(𝐸𝑎))−𝜈(𝜎^𝑚(𝑎))𝐸^{𝜎𝑖(𝑎),𝜆𝛼,𝑝−1}(𝜎^𝑚(𝑎))=𝐸^{𝜎𝑖(𝑎),𝜆𝛼,𝑝}𝜎^𝑚−1(𝑎),𝑖=0,…,𝑚−2,_{𝜎𝑖(𝑎),𝜆𝛼,𝑝}(𝜎^𝑚(𝑎))−𝜈(𝜎^𝑚(𝑎))𝐸^{𝜎𝑖(𝑎),𝜆𝛼,𝑝−1}(𝜎^𝑚(𝑎))=0,𝑖=𝑚−1,(4.22) where 𝑝∈ℤ,𝑝≥3−𝑚+𝑖. Starting with the last column, using 𝑚−1 elementary column operations of the type (4.22), we obtain the matrix ⎛⎜⎜⎜⎜⎜⎝𝐸𝐸^{𝜎𝑚−1(𝑎),𝜆𝛼,1}(𝜎^𝑚(𝑎))_{𝜎𝑚−2(𝑎),𝜆𝛼,0}𝐸⋮(𝜎^𝑚(𝑎))_{𝑎,𝜆𝛼,2−𝑚}(𝜎^𝑚(𝑎))0⋯0𝑀_𝑚−1𝜎^𝑚−1(𝑎)⎞⎟⎟⎟⎟⎟⎠(4.23) preserving the value of det𝑀_𝑚(𝜎^𝑚(𝑎)). Since 𝐸^{𝜎𝑚−1(𝑎),𝜆𝛼,1}(𝜎^𝑚(𝑎))=^∞_𝑘=0𝜆^𝑘(𝜈(𝜎^𝑚(𝑎)))^𝛼𝑘=11−𝜆(𝜈(𝜎^𝑚(𝑎)))^𝛼,(4.24) we can observe the recurrence det𝑀_𝑚(𝜎^𝑚(𝑎))=11−𝜆(𝜎^𝑚(𝑎))^𝛼det𝑀_𝑚−1𝜎^𝑚−1(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 [𝜎(𝑎),𝑏]∩𝕋^{𝜎(𝑎)(𝑞,ℎ)}, where 𝑏∈𝕋^{𝜎(𝑎)(𝑞,ℎ)} is satisfying |𝜆|(𝜈(𝑏))^𝛼<1. Then 𝑦(𝑡)=^𝑚_𝑗=1𝑐_𝑗𝐸^{𝑎,𝜆𝛼,𝛼−𝑚+𝑗}(𝑡),(4.26) where 𝑐₁,…,𝑐_𝑚 are real constants.

We conclude this paper by the illustrating example.

Example 4.8. Consider the initial value problem _𝑎∇^{𝛼(𝑞,ℎ)}𝑦(𝑡)=𝜆𝑦(𝑡),𝜎³(𝑎)≤𝑡≤𝜎^𝑛(𝑎),1<𝛼≤2,_𝑎∇^{𝛼−1(𝑞,ℎ)}𝑦(𝑡)||_{𝑡=𝜎2(𝑎)}=𝑦_𝛼−1,_𝑎∇^{𝛼−2(𝑞,ℎ)}𝑦(𝑡)||_{𝑡=𝜎2(𝑎)}=𝑦_𝛼−2,(4.27) 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)=𝑐₁𝐸^{𝑎,𝜆𝛼,𝛼−1}(𝑡)+𝑐₂𝐸^{𝑎,𝜆𝛼,𝛼}(𝑡).(4.28) The constants 𝑐₁, 𝑐₂ can be determined from the system 𝑉_2−𝛼𝐸^{𝑎,𝜆𝛼,𝛼−1},𝐸^{𝑎,𝜆𝛼,𝛼}𝜎²(𝑎)⋅𝑐¹𝑐₂=𝑦𝑦^𝛼−2_𝛼−1(4.29) with the matrix elements 𝑣₁₁=𝑣₂₂=[1]^𝑞+[𝛼]_𝑞−[1]_𝑞𝜆𝜈(𝜎(𝑎))^𝛼𝑣₁₂(1−𝜆𝜈(𝜎(𝑎))^𝛼)1−𝜆𝜈𝜎²(𝑎)^𝛼,=[2]^𝑞𝜈(𝜎(𝑎))+[𝛼]_𝑞−[2]_𝑞𝜆𝜈(𝜎(𝑎))^𝛼+1𝑣₂₁(1−𝜆𝜈(𝜎(𝑎))^𝛼)1−𝜆𝜈𝜎²(𝑎)^𝛼,=[𝛼]^𝑞𝜆𝜈(𝜎(𝑎))^𝛼−1(1−𝜆𝜈(𝜎(𝑎))^𝛼)1−𝜆𝜈𝜎²(𝑎)^𝛼.(4.30) By Lemma 4.6, the matrix 𝑉_2−𝛼(𝐸^{𝑎,𝜆𝛼,𝛼−1},𝐸^{𝑎,𝜆𝛼,𝛼})(𝜎²(𝑎)) has a nonzero determinant, hence applying the Cramer rule, we get 𝑐₁=𝑦^𝛼−2𝑣₂₂−𝑦_𝛼−1𝑣₁₂𝑐𝑊_2−𝛼𝐸^{𝑎,𝜆𝛼,𝛼−1},𝐸^{𝑎,𝜆𝛼,𝛼}𝜎²(𝑎)₂,=𝑦^𝛼−1𝑣₁₁−𝑦_𝛼−2𝑣₂₁𝑊_2−𝛼𝐸^{𝑎,𝜆𝛼,𝛼−1},𝐸^{𝑎,𝜆𝛼,𝛼}𝜎²(𝑎).(4.31)
Now we make a particular choice of the parameters 𝛼, 𝑎, 𝜆, 𝑦_𝛼−1 and 𝑦_𝛼−2 and consider the initial value problem in the form ₁∇^{1.8(𝑞,ℎ)}𝑦(𝑡)=−13𝑦(𝑡),𝜎³(1)≤𝑡≤𝜎^𝑛(1),₁∇^{0.8(𝑞,ℎ)}𝑦(𝑡)|||_{𝑡=𝜎2(1)}=−1,₁∇^{−0.2(𝑞,ℎ)}𝑦(𝑡)|||_{𝑡=𝜎2(1)}=1,(4.32) where 𝑛 is a positive integer satisfying 𝜈(𝜎^𝑛(1))<3^5/9. If we take the time scale of integers (the case 𝑞=ℎ=1), then the solution 𝑦(𝑡) of the corresponding initial value problem takes the form 𝑦(𝑡)=145^∞_𝑘=0−13^𝑘∏^{𝑡−2𝑗=1}(𝑗+1.8𝑘−0.2)(𝑡−2)!−125^∞_𝑘=0−13^𝑘∏^{𝑡−2𝑗=1}(𝑗+1.8𝑘+0.8)(𝑡−2)!,𝑡=2,3,….(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.

Figure 1 

𝛼=1.8,𝑎=1,𝜆=−1/3,𝑦𝛼−1=−1,𝑦𝛼−2=1.

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.