A Generalized q-Mittag-Leffler Function by q-Captuo Fractional Linear Equations

Abstract

Some Caputo q-fractional difference equations are solved. The solutions are expressed by means of a new introduced generalized type of q-Mittag-Leffler functions. The method of successive approximation is used to obtain the solutions. The obtained q-version of Mittag-Leffler function is thought as the q-analogue of the one introduced previously by Kilbas and Saigo (1995).

1. Introduction and Preliminaries

The concept of fractional calculus is not new. However, it has gained its popularity and importance during the last three decades or so. This is due to its distinguished applications in numerous diverse fields of science and engineering (see, e.g., [1–6] and the references therein). The q-calculus is also not of recent appearance. It was initiated in the twenties of the last century. For the basic concepts in q-calculus we refer the reader to [7]. Discrete and q-fractional difference equations are discrete versions of fractional differential equations. An extensive work has been done in discrete fractional dynamic equations and discrete fractional variational calculus (see [8–12]). Some of the authors applied the delta analysis and some applied nabla analysis. Since the domain of nabla operatos is more stable, the nabla approach could be preferable. In this paper we apply the nabla approach in the quantum case with 0<𝑞<1, but also the delta approach is possible [13]. During the last decade many authors applied diverse methods, such as homotopy perturbation method, to derive approximate analytical solutions of systems of fractional differential equations into Caputo and Riemann (see [14–18]). In this paper, we apply a direct method to express the solution of a certain linear Caputo q-fractional differential equation by means of a new introduced generalized q-Mittag-Leffler function.

Starting from the q-analogue of Cauchy formula [19], Al-Salam started the fitting of the concept of q-fractional calculus. After that he [20, 21] and Agarwal [22] continued on by studying certain q-fractional integrals and derivatives, where they proved the semigroup properties for left and right (Riemann) type fractional integrals but without variable lower limit and variable upper limit, respectively. Recently, the authors in [23] generalized the notion of the (left) fractional q-integral and q-derivative by introducing variable lower limit and proved the semigroup properties.

Very recently and after the appearance of time-scale calculus (see, e.g., [24]), some authors started to pay attention and apply the techniques of time scale to discrete fractional calculus (see [25–28]) benefitting from the results announced before in [29]. All of these results are mainly about fractional calculus on the time scales 𝕋_𝑞={𝑞^𝑛∶𝑛∈ℤ}∪{0} and ℎℤ [30]. As a contribution in this direction and being motivated by what is mentioned before, in this paper we introduce the q-analogue of a generalized type Mittag-Leffler function used before by Kilbas and Saigo in [31]. Such functions are obtained by solving linear q-Caputo initial value problems. The results obtained in this paper generalize also the results of [32]. Indeed, the authors in [32] solved a linear Caputo q-fractional difference equation of the form_𝑞𝐶^𝛼_𝑎𝑦(𝑥)=𝜆𝑦(𝑥)+𝑓(𝑥),𝑦(𝑎)=𝑏,0<𝛼<1, (1.1) where the solution was expressed by means of discrete q-Mittag-Leffler functions. In this paper, we solve an equation of the form_𝑞𝐶^𝛼_𝑎𝑦(𝑥)=𝜆(𝑥−𝑎)^𝛽_𝑞𝑦𝑞^−𝛽𝑥,𝑦(𝑎)=𝑏,0<𝛼<1,𝛽>−𝛼,𝜆∈ℝ,𝑏∈ℝ, (1.2) where the solution is expressed by means of a more general discrete q-Mittag-Leffler functions generalizing the ones obtained by (1.1), as (1.1) is obtained from (1.2) by setting 𝛽=0. Finally, we generalize to the higher-order case for any 𝛼>0, where higher-order q-Mittag-Leffler functions are obtained.

For the theory of q-calculus we refer the reader to the survey of [7], and for the basic definitions and results for the q-fractional calculus we refer to [28]. Here we will summarize some of those basics.

For 0<𝑞<1, let 𝕋_𝑞 be the time scale: 𝕋_𝑞=𝑞^𝑛∶𝑛∈ℤ∪{0}, (1.3) where ℤ is the set of integers. More generally, if 𝛼 is a nonnegative real number, then we define the time scale𝕋^𝛼_𝑞=𝑞^𝑛+𝛼∶𝑛∈ℤ∪{0}, (1.4) and we write 𝕋⁰_𝑞=𝕋_𝑞.

For a function 𝑓∶𝑇_𝑞→ℝ, the nabla q-derivative of 𝑓 is given by∇_𝑞𝑓(𝑡)=𝑓(𝑡)−𝑓𝑞𝑡1−𝑞𝑡,𝑡∈𝕋_𝑞−{0}. (1.5) The nabla q-integral of 𝑓 is given by^𝑡₀𝑓(𝑠)∇_𝑞𝑠=1−𝑞𝑡^∞_𝑖=0𝑞^𝑖𝑓𝑡𝑞^𝑖, (1.6) and for 0≤𝑎∈𝑇_𝑞,^𝑡_𝑎𝑓(𝑠)∇_𝑞𝑠=^𝑡₀𝑓(𝑠)∇_𝑞𝑠−^𝑎₀𝑓(𝑠)∇_𝑞𝑠. (1.7) On the other hand^∞_𝑡𝑓(𝑠)∇_𝑞𝑠=1−𝑞𝑡^∞_𝑖=1𝑞^−𝑖𝑓𝑡𝑞^−𝑖, (1.8) and for 0<𝑏<∞ in 𝕋_𝑞,^𝑏_𝑡𝑓(𝑠)∇_𝑞𝑠=^∞_𝑡𝑓(𝑠)∇_𝑞𝑠−^∞_𝑏𝑓(𝑠)∇_𝑞𝑠. (1.9) By the fundamental theorem in q-calculus we have∇_𝑞^𝑡₀𝑓(𝑠)∇_𝑞𝑠=𝑓(𝑡), (1.10) and if 𝑓 is continuous at 0, then^𝑡₀∇_𝑞𝑓(𝑠)∇_𝑞𝑠=𝑓(𝑡)−𝑓(0). (1.11) Also the following identity will be helpful:∇_𝑞^𝑡_𝑎𝑓(𝑡,𝑠)∇_𝑞𝑠=^𝑡_𝑎∇_𝑞𝑓(𝑡,𝑠)∇_𝑞𝑠+𝑓𝑞𝑡,𝑡. (1.12) Similarly the following identity will be useful as well:∇_𝑞^𝑏_𝑡𝑓(𝑡,𝑠)∇_𝑞𝑠=^𝑏_𝑞𝑡∇_𝑞𝑓(𝑡,𝑠)∇_𝑞𝑠−𝑓(𝑡,𝑡). (1.13) The q-derivative in (1.12) and (1.13) is applied with respect to t.

From the theory of q-calculus and the theory of time scale more generally, the following product rule is valid:∇_𝑞𝑓(𝑡)𝑔(𝑡)=𝑓𝑞𝑡∇_𝑞𝑔(𝑡)+∇_𝑞𝑓(𝑡)𝑔(𝑡). (1.14) The q-factorial function for 𝑛∈ℕ is defined by(𝑡−𝑠)^𝑛_𝑞=^𝑛−1_𝑖=0𝑡−𝑞^𝑖𝑠. (1.15) When 𝛼 is a nonpositive integer, the q-factorial function is defined by(𝑡−𝑠)^𝛼_𝑞=𝑡^𝛼∞_𝑖=0(1−(𝑠/𝑡))𝑞^𝑖(1−(𝑠/𝑡))𝑞^𝑖+𝛼. (1.16) We summarize some of the properties of q-factorial functions, which can be found mainly in [28], in the following lemma.

Lemma 1.1. One has the following.
(i)(𝑡−𝑠)_𝑞^𝛽+𝛾=(𝑡−𝑠)^𝛽_𝑞(𝑡−𝑞^𝛽𝑠)^𝛾_𝑞.(ii)(𝑎𝑡−𝑎𝑠)^𝛽_𝑞=𝑎^𝛽(𝑡−𝑠)^𝛽_𝑞.(iii)The nabla q- derivative of the q- factorial function with respect to 𝑡 is∇_𝑞(𝑡−𝑠)^𝛼_𝑞=1−𝑞^𝛼()1−𝑞𝑡−𝑠_𝑞^𝛼−1. (1.17)(iv)The nabla q- derivative of the q- factorial function with respect to 𝑠 is ∇_𝑞(𝑡−𝑠)^𝛼_𝑞=−1−𝑞^𝛼1−𝑞𝑡−𝑞𝑠_𝑞^𝛼−1, (1.18) where 𝛼,𝛾,𝛽∈ℝ.

Definition 1.2 (see [32]). Let 𝛼>0. If 𝛼∉ℕ, then the 𝛼-order Caputo (left) q-fractional derivative of a function 𝑓 is defined by_𝑞𝐶^𝛼_𝑎𝑓(𝑡)≜_𝑞𝐼_𝑎^{(𝑛−𝛼)}∇^𝑛_𝑞𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡_𝑎𝑡−𝑞𝑠_𝑞^{𝑛−𝛼−1}∇^𝑛_𝑞𝑓(𝑠)∇_𝑞𝑠, (1.19) where 𝑛=[𝛼]+1.
If 𝛼∈ℕ, then _𝑞𝐶^𝛼_𝑎𝑓(𝑡)≜∇^𝑛_𝑞𝑓(𝑡).

It is clear that _𝑞𝐶^𝛼_𝑎 maps functions defined on 𝑇_𝑞 to functions defined on 𝑇_𝑞, and that _𝑏𝐶^𝛼_𝑞 maps functions defined on 𝑇_𝑞^1−𝛼 to functions defined on 𝑇_𝑞.

The following identity which is useful to transform Caputo q-fractional difference equations into q-fractional integrals, will be our key in solving the q-fractional linear type equation by using successive approximation.

Proposition 1.3 ([32]). Assume that 𝛼>0 and 𝑓 is defined in suitable domains. Then _𝑞𝐼^𝛼_𝑎𝑞𝐶^𝛼_𝑎𝑓(𝑡)=𝑓(𝑡)−^𝑛−1_𝑘=0(𝑡−𝑎)^𝑘_𝑞Γ_𝑞()∇𝑘+1^𝑘𝑞𝑓(𝑎), (1.20) and if 0<𝛼≤1, then _𝑞𝐼^𝛼_𝑎𝑞𝐶^𝛼_𝑎𝑓(𝑡)=𝑓(𝑡)−𝑓(𝑎). (1.21)

The following identity [23] is essential to solve linear q-fractional equations:_𝑞𝐼^𝛼_𝑎(𝑥−𝑎)^𝜇_𝑞=Γ_𝑞𝜇+1Γ_𝑞()𝛼+𝜇+1𝑥−𝑎_𝑞^𝜇+𝛼(0<𝑎<𝑥<𝑏), (1.22) where 𝛼∈ℝ⁺ and 𝜇∈(−1,∞). The q-analogue of Mittag-Leffler function with double index (𝛼,𝛽) is introduced in [32]. It was defined as follows.

Definition 1.4 ([32]). For 𝑧,𝑧₀∈𝐂 and ℜ(𝛼)>0, the q-Mittag-Leffler function is defined by _𝑞𝐸_𝛼,𝛽𝜆,𝑧−𝑧₀=^∞_𝑘=0𝜆^𝑘𝑧−𝑧_0𝑞^𝛼𝑘Γ_𝑞𝛼𝑘+𝛽. (1.23) When 𝛽=1, we simply use _𝑞𝐸_𝛼(𝜆,𝑧−𝑧₀)∶=_𝑞𝐸_𝛼,1(𝜆,𝑧−𝑧₀).

2. Main Results

The following is to be the q-analogue of the generalized Mittag-Leffler function introduced by Kilbas and Saigo [31] (see also [3] page 48).

Definition 2.1. For 𝛼,𝑙,𝜆∈ℂ are complex numbers and 𝑚∈ℝ such that ℜ(𝛼)>0,𝑚>0,𝑎≥0, and 𝛼(𝑗𝑚+𝑙)≠−1,−2,−3,…, the generalized q-Mittag-Leffler function (of order 0) is defined by _𝑞𝐸_{𝛼,𝑚,𝑙}(𝜆,𝑥−𝑎)=1+^∞_𝑘=1𝜆^𝑘𝑞^{−(𝑘(𝑘−1)/2)𝛼(𝑚−1)(𝛼𝑙+𝛼)}𝑐_𝑘(𝑥−𝑎)_𝑞^𝛼𝑘𝑚, (2.1) where 𝑐_𝑘=^𝑘−1_𝑗=0Γ_𝑞𝛼𝑗𝑚+𝑙+1Γ_𝑞𝛼𝑗𝑚+𝑙+1+1,𝑘=1,2,3,…, (2.2) while the generalized q-Mittag-Leffler function (of order r), 𝑟=0,1,2,3,…, is defined by _𝑞𝐸^𝑟_{𝛼,𝑚,𝑙}(𝜆,𝑥−𝑎)=1+^∞_𝑘=1𝜆^𝑘𝑞^{−𝑘𝛼(𝑚−1)𝑟}𝑞^{−(𝑘(𝑘−1)/2)𝛼(𝑚−1)(𝛼𝑙+𝛼)}𝑐_𝑘𝑥−𝑞^𝑟𝑎_𝑞^𝛼𝑘𝑚. (2.3) Note that _𝑞𝐸⁰_{𝛼,𝑚,𝑙}(𝜆,𝑥−𝑎)=_𝑞𝐸_{𝛼,𝑚,𝑙}(𝜆,𝑥−𝑎).

Remark 2.2. In particular, if 𝑚=1, then the generalized q-Mittag-Leffler function is reduced to the q-Mittag-Leffler function, apart from a constant factor Γ_𝑞(𝛼𝑙+1). Namely, _𝑞𝐸_𝛼,1,𝑙(𝜆,𝑥−𝑎)=Γ_𝑞(𝛼𝑙+1)_𝑞𝐸_{𝛼,𝛼𝑙+1}(𝜆,𝑥−𝑎). (2.4)
This turns to be the q-analogue of the identity 𝐸_𝛼,1,𝑙(𝑧)=Γ(𝛼𝑙+1)𝐸_{𝛼,𝛼𝑙+1}(𝑧) (see [3] page 48).

Example 2.3. Consider the q-Caputo difference equation: _𝑞𝐶^𝛼_𝑎𝑦(𝑥)=𝜆(𝑥−𝑎)^𝛽_𝑞𝑦𝑞^−𝛽𝑥,𝑦(𝑎)=𝑏,0<𝛼<1,𝛽>−𝛼,𝜆∈ℝ,𝑏∈ℝ. (2.5)
Applying Proposition 1.3 we have 𝑦(𝑥)=𝑦(𝑎)+𝜆_𝑞𝐼^𝛼_𝑎(𝑥−𝑎)^𝛽_𝑞𝑦𝑞^−𝛽𝑥. (2.6) The method of successive applications implies that 𝑦_𝑚(𝑥)=𝑦(𝑎)+𝜆_𝑞𝐼^𝛼_𝑎(𝑥−𝑎)^𝛽_𝑞𝑦_𝑚−1𝑞^−𝛽𝑥,𝑚=1,2,3,…, (2.7) where 𝑦₀(𝑥)=𝑏. Then by the help of (1.22) we have 𝑦₁(𝑥)Γ=𝑏+𝑏𝜆^𝑞𝛽+1Γ_𝑞()𝛽+𝛼+1𝑥−𝑎_𝑞^𝛽+𝛼,𝑦₂(𝑥)=𝑏+𝑏𝜆_𝑞𝐼^𝛼_𝑎(𝑥−𝑎)^𝛽_𝑞Γ1+𝜆^𝑞𝛽+1Γ_𝑞𝑞𝛽+𝛼+1^−𝛽𝑥−𝑎_𝑞^𝛽+𝛼. (2.8) Then by (i) and (ii) of Lemma 1.1, 𝑦₂(𝑥)=𝑏+𝑏𝜆_𝑞𝐼^𝛼_𝑎(𝑥−𝑎)^𝛽_𝑞Γ+𝜆^𝑞𝛽+1Γ_𝑞𝑞𝛽+𝛼+1^{−𝛽(𝛼+𝛽)}(𝑥−𝑎)_𝑞^2𝛽+𝛼. (2.9) Again by (1.22) we conclude that 𝑦₂(𝑥)=𝑏+𝑏𝜆_𝑞𝐼^𝛼_𝑎(𝑥−𝑎)^𝛽_𝑞Γ+𝜆^𝑞𝛽+1Γ_𝑞𝑞𝛽+𝛼+1^{−𝛽(𝛼+𝛽)}(𝑥−𝑎)_𝑞^2𝛽+𝛼. (2.10) Then (1.22) leads to 𝑦₂(𝑥)Γ=𝑏1+𝜆^𝑞𝛽+1Γ_𝑞()𝛽+𝛼+1𝑥−𝑎_𝑞^𝛽+𝛼+𝜆²Γ_𝑞2𝛽+𝛼+1Γ_𝑞𝑞2𝛽+2𝛼+1^{−𝛽(𝛼+𝛽)}(𝑥−𝑎)_𝑞^2𝛽+2𝛼. (2.11) Proceeding inductively, for each 𝑚=1,2,… we obtain 𝑦_𝑚(𝑥)=𝑏1+^𝑚_𝑘=1𝜆^𝑘𝑞^{−𝛽(𝑘(𝑘−1)/2)(𝛼+𝛽)}𝑐_𝑘(𝑥−𝑎)_𝑞^{𝑘(𝛼+𝛽)}, (2.12) where 𝑐_𝑘=^𝑘−1_𝑗=0Γ_𝑞𝛼𝑗𝑚+𝑙+1Γ_𝑞𝛼𝛽𝑗𝑚+𝑙+1+1,𝑚=1+𝛼𝛽,𝑙=𝛼,𝑘=1,2,3,…. (2.13)

If we let 𝑚→∞, then we obtain the solution𝑦(𝑥)=𝑏1+^∞_𝑘=1𝜆^𝑘𝑞^{−𝛽(𝑘(𝑘−1)/2)(𝛼+𝛽)}𝑐_𝑘(𝑥−𝑎)_𝑞^{𝑘(𝛼+𝛽)}. (2.14)

Now, by means of Definition 2.1, we can state the following.

Theorem 2.4. The solution of the q-Caputo difference equation (2.5) is given by 𝑦(𝑥)=𝑏_𝑞𝐸_{𝛼,(1+(𝛽/𝛼)),𝛽/𝛼}(𝜆,𝑥−𝑎). (2.15)

Remark 2.5. (1) If in (2.5) 𝛽=0, then in accordance with (2.4) and Example  9 in [32] we have _𝑞𝐸_𝛼,1,0(𝜆,𝑥−𝑎)=_𝑞𝐸_𝛼,1(𝜆,𝑥−𝑎)=_𝑞𝐸_𝛼(𝜆,𝑥−𝑎). (2.16)
(2) The solution of the q-Cauchy problem _𝑞𝐶_𝑎^1/2𝑦(𝑥)=𝜆(𝑥−𝑎)^𝛽_𝑞𝑦𝑞^−𝛽𝑥(𝑎)1,𝑦=𝑏,0<𝛼<1,𝛽>−2,𝜆∈ℝ,𝑏∈ℝ, (2.17) is given by 𝑦(𝑥)=𝑏_𝑞𝐸_{1/2,1+2𝛽,2𝛽}(𝜆,𝑥−𝑎). (2.18)

For the sake of generalization to the higher-order case, we consider the fractional q-initial value problem:_𝑞𝐶^𝛼_𝑎𝑦(𝑥)=𝜆(𝑥−𝑎)^𝛽_𝑞𝑦𝑞^−𝛽𝑥,𝑦^(𝑘)(𝑎)=𝑏_𝑘𝑏_𝑘∈ℝ,𝑘=0,1,…,𝑛−1, (2.19) where 𝑛−1<𝛼<𝑛,𝛽>−𝛼,𝜆∈ℝ,𝑏∈ℝ. (2.20)

Theorem 2.6. The solution of the fractional q- initial value problem (2.19) is of the following form: 𝑦(𝑥)=^𝑛−1_𝑟=0𝑏_𝑟Γ_𝑞()()𝑟+1𝑥−𝑎^𝑟_𝑞𝑞𝐸^𝑟_{,𝛼,1+𝛽/𝛼𝛽+𝑟/𝛼}(𝜆,𝑥−𝑎). (2.21)

Proof. The proof follows by the help of (1.20) and let Lemma 1.1 and by applying the successive approximation with 𝑦₀(𝑥)=^𝑛−1_𝑘=0(𝑡−𝑎)^𝑘_𝑞Γ_𝑞()∇𝑘+1^𝑘𝑞𝑓(𝑎), (2.22)

Note that when 0<𝛼<1, that is, 𝑛=1, the solution of Example 2.3 is recovered.

Next, we solve a nonhomogenous versions of (2.5).

Lemma 2.7. Let 𝑟∈ℕ, 𝛼>0, and let 𝑓 be defined on 𝕋_𝑞. Then _𝑞𝐼_𝑎𝑓𝑞^−𝑟𝑡=𝑞^𝑟𝛼_𝑞𝐼_{𝑞−𝑟𝑎}𝑓𝑞^−𝑟𝑡∀𝑡∈𝕋_𝑞. (2.23) In particular, if 𝑎=0, then _𝑞𝐼₀𝑓𝑞^−𝑟𝑡=𝑞^𝑟𝛼_𝑞𝐼₀𝑓𝑞^−𝑟𝑡∀𝑡∈𝕋_𝑞. (2.24)

Proof. The proof can be achieved by making use of Theorem  1 in [28] for integration by substitution (for details see [24]). Indeed, _𝑞𝐼_𝑎𝑓𝑞^−𝑟𝑡=1Γ_𝑞(𝛼)^𝑡_𝑎𝑡−𝑞𝑠_𝑞^𝛼−1𝑓𝑞^−𝑟𝑠∇_𝑞𝑠=𝑞^𝑟Γ_𝑞(𝛼)^{𝑞−𝑟𝑡}_𝑞^−𝑟_𝑎𝑡−𝑞𝑞^𝑟𝑠_𝑞^𝛼−1𝑓(𝑠)∇_𝑞𝑠=𝑞^𝑟𝛼Γ_𝑞(𝛼)^{𝑞−𝑟𝑡}_𝑞^−𝑟_𝑎𝑞^−𝑟𝑡−𝑞𝑠_𝑞^𝛼−1𝑓(𝑠)∇_𝑞𝑠=𝑞^𝑟𝛼_𝑞𝐼_{𝑞−𝑟𝑎}𝑓𝑞^−𝑟𝑡. (2.25)

Consider the 𝑞-fractional initial value problem:_𝑞𝐶^𝛼₀𝑦(𝑥)=𝜆𝑥^𝛽𝑦𝑞^−𝛽𝑥+𝑓(𝑥),𝑦(0)=𝑏, (2.26) where0<𝛼<1,𝛽>−𝛼,𝛽∈ℕ₀,𝜆∈ℝ,𝑏∈ℝ. (2.27) If we apply the successive approximation as in Example 2.3 and use Lemma 2.7, then we can state the following

Theorem 2.8. The solution of the initial value problem (2.26) is expressed by 𝑦(𝑥)=𝑏_𝑞𝐸_{𝛼,(1+(𝛽/𝛼)),𝛽/𝛼}(𝜆,𝑥)+^∞_𝑘=0𝜆^𝑘Γ_𝑞()𝑞𝛼𝑘+𝛼^{−𝛼𝛽(𝑘(𝑘+1)/2)𝑥}₀𝑥−𝑞𝑡_𝑞^{𝛼𝑘+𝛼}𝑓𝑞^−𝑘𝛽𝑡∇_𝑞𝑡. (2.28)

Remark 2.9. If in (2.26) we set 𝛽=0, then Example  9 in [32] is recovered for a=0.

Definition 2.10. A function 𝑓∶𝕋_𝑞→ℝ is called periodic with period 𝛽∈ℕ₁ if 𝛽 is the smallest natural number such that 𝑓(𝑞^𝛽𝑡)=𝑓(𝑡), for all 𝑡∈𝕋_𝑞.
Consider the nonhomogeneous initial value problem: _𝑞𝐶^𝛼₀𝑦(𝑥)=𝜆(𝑥−𝑎)^𝛽_𝑞𝑦𝑞^−𝛽𝑥+𝑓(𝑥),𝑦(𝑎)=𝑏, (2.29) where 0<𝛼<1,𝛽>−𝛼,𝛽∈ℕ₀,𝜆∈ℝ,𝑏∈ℝ. (2.30) If we apply the successive approximation as in Example 2.3, then we state the following.

Theorem 2.11. If in (2.29) either 𝛽=0 or 𝑓 is periodic with period dividing 𝛽, then the solution is given by 𝑦(𝑥)=𝑏_𝑞𝐸_{𝛼,(1+(𝛽/𝛼)),𝛽/𝛼}(𝜆,𝑥−𝑎)+^𝑥_𝑎𝑥−𝑞𝑡_𝑞𝑞^𝛼−1𝐸_𝛼,𝛼𝜆,𝑥−𝑞^𝛼𝑡𝑓(𝑡)∇_𝑞𝑡. (2.31)

Clearly, if 𝛽=0, then the result in Example  9 in [32] is recovered as well.

For the sake of completeness, it would be interesting if the h-discrete fractional analogue, or more generally the (𝑞,ℎ)-analogue of the general q-Mittag-Leffler functions are obtained, possibly better, by applying nabla calculus (see [33–35]). However, this needs preparations in the Caputo case and it might be very complicated.