# A Generalizedq-Mittag-Leffler Function by q-Captuo Fractional Linear Equations

Authors: Thabet Abdeljawad; Betül Benli; Dumitru Baleanu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Some Caputoq-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).


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## Body

## 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 theq-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 𝕋q={qn: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(1.1)(Caαyq)(x)=λy(x)+f(x),y(a)=b,0<α<1,
where the solution was expressed by means of discrete q-Mittag-Leffler functions. In this paper, we solve an equation of the form(1.2)(Caαyq)(x)=λ(x-a)qβy(q-βx),y(a)=b,0<α<1,β>-α,λ∈R,b∈R,
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 ofq-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.For0<q<1, let 𝕋q be the time scale:
(1.3)Tq={qn:n∈Z}∪{0},
where ℤ is the set of integers. More generally, if α is a nonnegative real number, then we define the time scale(1.4)Tqα={qn+α:n∈Z}∪{0},
and we write 𝕋q0=𝕋q.For a functionf:Tq→ℝ, the nabla q-derivative of f is given by(1.5)∇qf(t)=f(t)-f(qt)(1-q)t,t∈Tq-{0}.
The nabla q-integral of f is given by(1.6)∫0tf(s)∇qs=(1-q)t∑i=0∞qif(tqi),
and for 0≤a∈Tq,(1.7)∫atf(s)∇qs=∫0tf(s)∇qs-∫0af(s)∇qs.
On the other hand(1.8)∫t∞f(s)∇qs=(1-q)t∑i=1∞q-if(tq-i),
and for 0<b<∞ in 𝕋q,(1.9)∫tbf(s)∇qs=∫t∞f(s)∇qs-∫b∞f(s)∇qs.
By the fundamental theorem in q-calculus we have(1.10)∇q∫0tf(s)∇qs=f(t),
and if f is continuous at 0, then(1.11)∫0t∇qf(s)∇qs=f(t)-f(0).
Also the following identity will be helpful:(1.12)∇q∫atf(t,s)∇qs=∫at∇qf(t,s)∇qs+f(qt,t).
Similarly the following identity will be useful as well:(1.13)∇q∫tbf(t,s)∇qs=∫qtb∇qf(t,s)∇qs-f(t,t).
The q-derivative in (1.12) and (1.13) is applied with respect to t.From the theory ofq-calculus and the theory of time scale more generally, the following product rule is valid:(1.14)∇q(f(t)g(t))=f(qt)∇qg(t)+∇qf(t)g(t).
The q-factorial function for n∈ℕ is defined by(1.15)(t-s)qn=∏i=0n-1(t-qis).
When α is a nonpositive integer, the q-factorial function is defined by(1.16)(t-s)qα=tα∏i=0∞(1-(s/t))qi(1-(s/t))qi+α.
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)qβ+γ=(t-s)qβ(t-qβs)qγ.(ii)
(at-as)qβ=aβ(t-s)qβ.(iii)
The nabla q- derivative of the q- factorial function with respect tot is(1.17)∇q(t-s)qα=1-qα1-q(t-s)qα-1.(iv)
The nabla q- derivative of the q- factorial function with respect tos is
(1.18)∇q(t-s)qα=-1-qα1-q(t-qs)qα-1,
where α,γ,β∈ℝ.Definition 1.2 (see [32]).
Letα>0. Ifα∉ℕ, then the α-order Caputo (left)q-fractional derivative of a function f is defined by(1.19)qCaαf(t)≜Iqa(n-α)∇qnf(t)=1Γ(n-α)∫at(t-qs)qn-α-1∇qnf(s)∇qs,
where n=[α]+1.
Ifα∈ℕ, then Cqaαf(t)≜∇qnf(t).It is clear thatCqaα maps functions defined on Tq to functions defined on Tq, and that Cbqα maps functions defined on Tq1-α to functions defined on Tq.The following identity which is useful to transform Caputoq-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 thatα>0 and  f is defined in suitable domains. Then
(1.20)qIaαCqaαf(t)=f(t)-∑k=0n-1(t-a)qkΓq(k+1)∇qkf(a),
and if 0<α≤1, then
(1.21)qIaαqCaαf(t)=f(t)-f(a).The following identity [23] is essential to solve linear q-fractional equations:(1.22)qIaα(x-a)qμ=Γq(μ+1)Γq(α+μ+1)(x-a)qμ+α(0<a<x<b),
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]).
Forz,z0∈C and ℜ(α)>0, the q-Mittag-Leffler function is defined by
(1.23)qEα,β(λ,z-z0)=∑k=0∞λk(z-z0)qαkΓq(αk+β).
When β=1, we simply use Eqα(λ,z-z0)∶=Eqα,1(λ,z-z0).


## 2. Main Results

The following is to be theq-analogue of the generalized Mittag-Leffler function introduced by Kilbas and Saigo [31] (see also [3] page 48).Definition 2.1.
Forα,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 generalizedq-Mittag-Leffler function (of order 0) is defined by
(2.1)qEα,m,l(λ,x-a)=1+∑k=1∞λkq-(k(k-1)/2)α(m-1)(αl+α)ck(x-a)qαkm,
where
(2.2)ck=∏j=0k-1Γq[α(jm+l)+1]Γq[α(jm+l+1)+1],k=1,2,3,…,
while the generalized q-Mittag-Leffler function (of order r), r=0,1,2,3,…, is defined by
(2.3)Eqα,m,lr(λ,x-a)=1+∑k=1∞λkq-kα(m-1)rq-(k(k-1)/2)α(m-1)(αl+α)ck(x-qra)qαkm.
Note that Eqα,m,l0(λ,x-a)=Eqα,m,l(λ,x-a).Remark 2.2.
In particular, ifm=1, then the generalized q-Mittag-Leffler function is reduced to the q-Mittag-Leffler function, apart from a constant factor Γq(αl+1). Namely,
(2.4)qEα,1,l(λ,x-a)=Γq(αl+1)qEα,αl+1(λ,x-a).
This turns to be theq-analogue of the identity Eα,1,l(z)=Γ(αl+1)Eα,αl+1(z) (see [3] page 48).Example 2.3.
Consider theq-Caputo difference equation:
(2.5)(Cqaαy)(x)=λ(x-a)qβy(q-βx),y(a)=b,0<α<1,β>-α,λ∈R,b∈R.
Applying Proposition1.3 we have
(2.6)y(x)=y(a)+λIqaα[(x-a)qβy(q-βx)].
The method of successive applications implies that
(2.7)ym(x)=y(a)+λIqaα[(x-a)qβym-1(q-βx)],m=1,2,3,…,
where y0(x)=b. Then by the help of (1.22) we have
(2.8)y1(x)=b+bλΓq(β+1)Γq(β+α+1)(x-a)qβ+α,y2(x)=b+bλIqaα[(x-a)qβ{1+λΓq(β+1)Γq(β+α+1)(q-βx-a)qβ+α}].
Then by (i) and (ii) of Lemma 1.1,
(2.9)y2(x)=b+bλIqaα[(x-a)qβ+λΓq(β+1)Γq(β+α+1)q-β(α+β)(x-a)q2β+α].
Again by (1.22) we conclude that
(2.10)y2(x)=b+bλIqaα[(x-a)qβ+λΓq(β+1)Γq(β+α+1)q-β(α+β)(x-a)q2β+α].
Then (1.22) leads to
(2.11)y2(x)=b[1+λΓq(β+1)Γq(β+α+1)(x-a)qβ+α+λ2Γq(2β+α+1)Γq(2β+2α+1)q-β(α+β)(x-a)q2β+2α].
Proceeding inductively, for each m=1,2,… we obtain
(2.12)ym(x)=b[1+∑k=1mλkq-β(k(k-1)/2)(α+β)ck(x-a)qk(α+β)],
where
(2.13)ck=∏j=0k-1Γq[α(jm+l)+1]Γq[α(jm+l+1)+1],m=1+βα,l=βα,k=1,2,3,….If we letm→∞, then we obtain the solution(2.14)y(x)=b[1+∑k=1∞λkq-β(k(k-1)/2)(α+β)ck(x-a)qk(α+β)].Now, by means of Definition2.1, we can state the following.Theorem 2.4.
The solution of the q-Caputo difference equation (2.5) is given by
(2.15)y(x)=bEqα,(1+(β/α)),β/α(λ,x-a).Remark 2.5.
(1) If in(2.5)β=0, then in accordance with (2.4) and Example  9 in [32] we have
(2.16)Eqα,1,0(λ,x-a)=Eqα,1(λ,x-a)=Eqα(λ,x-a).
(2) The solution of theq-Cauchy problem
(2.17)(Cqa1/2y)(x)=λ(x-a)qβy(q-βx),y(a)=b,0<α<1,β>-12,λ∈R,b∈R,
is given by
(2.18)y(x)=bEq1/2,1+2β,2β(λ,x-a).For the sake of generalization to the higher-order case, we consider the fractionalq-initial value problem:(2.19)(Cqaαy)(x)=λ(x-a)qβy(q-βx),y(k)(a)=bk(bk∈R,k=0,1,…,n-1),
where
(2.20)n-1<α<n,β>-α,λ∈R,b∈R.Theorem 2.6.
The solution of the fractional q- initial value problem (2.19) is of the following form:
(2.21)y(x)=∑r=0n-1brΓq(r+1)(x-a)qrqEα,((1+β)/α),((β+r)/α)r(λ,x-a).Proof.
The proof follows by the help of (1.20) and let Lemma 1.1 and by applying the successive approximation with
(2.22)y0(x)=∑k=0n-1(t-a)qkΓq(k+1)∇qkf(a),Note that when0<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.
Letr∈ℕ, α>0,  and let f be defined on  𝕋q. Then
(2.23)qIaf(q-rt)=qrα(qIq-raf)(q-rt)∀t∈Tq.
In particular, if a=0, then
(2.24)qI0f(q-rt)=qrα(qI0f)(q-rt)∀t∈Tq.Proof.
The proof can be achieved by making use of Theorem  1 in [28] for integration by substitution (for details see [24]). Indeed,
(2.25)qIaf(q-rt)=1Γq(α)∫at(t-qs)qα-1f(q-rs)∇qs=qrΓq(α)∫q-raq-rt(t-qqrs)qα-1f(s)∇qs=qrαΓq(α)∫q-raq-rt(q-rt-qs)qα-1f(s)∇qs=qrα(qIq-raf)(q-rt).Consider theq-fractional initial value problem:(2.26)(Cq0αy)(x)=λxβy(q-βx)+f(x),y(0)=b,
where(2.27)0<α<1,β>-α,β∈N0,λ∈R,b∈R.
If we apply the successive approximation as in Example 2.3 and use Lemma 2.7, then we can state the followingTheorem 2.8.
The solution of the initial value problem (2.26) is expressed by
(2.28)y(x)=bEqα,(1+(β/α)),β/α(λ,x)+∑k=0∞λkΓq(αk+α)q-αβ(k(k+1)/2)∫0x(x-qt)qαk+αf(q-kβt)∇qt.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 functionf:𝕋q→ℝ is called periodic with period β∈ℕ1ifβ is the smallest natural number such that f(qβt)=f(t), for all t∈𝕋q.
Consider the nonhomogeneous initial value problem:(2.29)(qC0αy)(x)=λ(x-a)qβy(q-βx)+f(x),y(a)=b,
where
(2.30)0<α<1,β>-α,β∈N0,λ∈R,b∈R.
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
(2.31)y(x)=bqEα,(1+(β/α)),β/α(λ,x-a)+∫ax(x-qt)qα-1qEα,α(λ,x-qαt)f(t)∇qt.Clearly, ifβ=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.



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