A Kilbas

D Baleanu

Dumitru Baleanu

Fahd Jarad

FM Atıcı

G Samko

GA Anastassiou

HL Gray

I Podlubny

KS Miller

M Bohner

Nuno RO Bastos

RL Magin

T Abdeljawad

Thabet Abdeljawad

Advances in Difference Equations

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RESEARCH Open AccessA semigroup-like Property for DiscreteMittag-Leffler FunctionsThabet Abdeljawad1*, Fahd Jarad1 and Dumitru Baleanu1,2* Correspondence:thabet@cankaya.edu.tr1Department of Mathematics,Çankaya University, 06530, Ankara,TurkeyFull list of author information isavailable at the end of the articleAbstractDiscrete Mittag-Leffler function Eα¯(λ, z) of order 0 <a ≤ 1, E1¯(λ, z) = (1 − λ)−z, l ≠1, satisfies the nabla Caputo fractional linear difference equationC∇α0 x(t) = λx(t), x(0) = 1, t ∈ N1 = {1, 2, 3, . . .}.Computations can show that the semigroup identityEα¯(λ, z1)Eα¯(λ, z2) = Eα¯(λ, z1 + z2)does not hold unless l = 0 or a = 1. In this article we develop a semigroupproperty for the discrete Mittag-Leffler function Eα¯(λ, z) in the case a ↑ 1 is just theabove identity. The obtained semigroup identity will be useful to develop anoperator theory for the discrete fractional Cauchy problem with order a Î (0, 1).Keywords: Caputo fractional difference, discrete Mittag-Leffler function, discretenabla Laplace transform, convolution1 Introduction and PreliminariesThe fractional calculus started to be investigated deeply in both theorical and appliedview points. One of the main reasons of the fast development of this type of calculusis that it incorporates, as a particular case, the classical calculus. Starting from thisinteresting point it is natural to ask if there is an extension of the fractional calculusand even more precisely if this generalization could find us new dimensions of someproblems within the complex systems which up to now were not solved properly as inBiology, nanotechnology and medicine [1-6]. Mittag-Leffler functions play a veryimportant role in the theory of fractional differential equations [1-4].Recently there is a huge effort on the line of discretizing the fractional calculusoperators and its applications in the control theory and the corresponding variationalprinciples [9-17]. In our opinion the discrete fractional operators can play a crucialrole both from the theoretical and applied point of view. However the combinations ofthe techniques from both fields are not always straithforward namely because the frac-tional operators are non-local. However, we expect that this new unification will pro-vide new tools in understanding the hyper-complex dynamical systems.In this article, the main aim is to establish a semi-group property for discrete Mittag-Leffler functions.This article is organized as follows: The rest of this section contains definitions andpreliminary concepts regarding the rising factorial function, the discrete Mittaf-LefflerAbdeljawad et al. Advances in Difference Equations 2012, 2012:72http://www.advancesindifferenceequations.com/content/2012/1/72© 2012 Abdeljawad et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.functions, Caputo fractional difference and the discrete convolution. In Section 2, wefind the nabla discrete transforms of certain discrete Mittag-Leffler functions by mak-ing use of the discrete convolution theorem, which will be helpful to proceed inobtaining our main results. Finally, Section 3 deals with a semigroup property for dis-crete Mittag-Leffler functions and some examples are given to illustrate our results.For the sake of the nabla fractional calculus we have the following definitionDefinition 1. (i) For a natural number m, the m rising (ascending) factorial of t isdefined bytm¯ =m−1∏k=0(t + k), t0¯ = 1. (1)(ii) For any real number the a rising function is defined bytα¯ =(t + α)(t), t ∈ R\{. . . ,−2,−1, 0}, 0α¯ = 0 (2)Regarding the rising factorial function we observe the following:(i)∇(tα¯) = αtα−1 (3)(ii)∇t(s − t)α¯ = α(s − ρ(t))α−1, (4)where r(t) = t - 1.Definition 2. The nabla discrete Laplace transform for a function f defined on N0isdefined byN f (z) =∞∑t=1(1 − z)t−1f (t) (5)If f(t, s) is a function of two variables, we state explicitly to specify to which para-meter we apply the transform.Lemma 1. [11]For any α ∈ R\{. . . ,−2,−1, 0},• (i) N (tα−1)(z) = (α)sα , |1 - z| < 1,• (ii) N (tα−1b−t)(z) = bα−1(α)(s+b−1)α , |1 - z| <b.Definition 3. (Nabla Discrete Mittag-Leffler) For λ ∈ Rand a, b, z ∈ Cwith Re(a) >0,the nabla discrete Mittag-Leffler functions are defined byEα,β(λ, z) =∞∑k=0λkzkα+β−1(αk + β). (6)For b = 1, it is written thatEα¯(λ, z)  Eα,1(λ, z) =∞∑k=0λkzkα(αk + 1). (7)Abdeljawad et al. Advances in Difference Equations 2012, 2012:72http://www.advancesindifferenceequations.com/content/2012/1/72Page 2 of 7Definition 4. For 0 < a ≤ 1 the a-order Caputo fractional derivative of a function f,which is defined on N0, is defined byC∇α0 f (t) = ∇−(1−α)0 ∇f (t)=1(1 − α)t∑s=1(t − ρ(s))−α∇f (s),where r(s) = s - 1 and ∇−α0 f (t) = 1(α)∑ts=1 (t − ρ(s))α−1f (s)is the nabla left frac-tional sum of order a.Note that if f is defined on N0, then C∇α0 f (t) is defined on N1 = {1, 2, 3, . . .}.Example 1. Let 0 < a ≤ 1, a ∈ Rand consider the nabla left Caputo nonhomogeneousfractional difference equationC∇α0 y(t) = λy(t) + f (t), y(0) = a0, t ∈ N0. (8)The solution of (8) has the formy(t) = a0Eα¯(λ, t) +t∑s=1Eα,α(λ, t − ρ(s))f (s). (9)Remark 1. If we solve the nabla discrete fractional system (8) with a = 1 and a0 = 1we obtain the following solutiony(t) =∞∑k=0λktk¯k!+t∑s=1∞∑k=0λk(t − ρ(s))k¯k!f (s).The first part of the solution is the nabla discrete exponential functioneλ(t, 0) = (1 − λ)−t. For the sake of more comparisons see reference ([18], p. 118).Definition 5. Let s ∈ R, 0 < a <1 and f : N0 → Rbe a function. The nabla discreteconvolution of gαs (t) = (t + s)α¯with f is defined by(gαs ∗ f )(t) =t∑τ=1(t + s − ρ(τ ))α¯f (τ ). (10)2 Discrete Laplace of Mittag-Leffler Type and Convolution Type FunctionsProposition 1. For any α ∈ R\{. . . ,−2,−1, 0}, s ∈ Rand f defined on N0we have(N (gαs ∗ f ))(z) = (N gαs )(z)(N f )(z). (11)Proof.(N (gαs ∗ f ))(z) =∞∑t=1(1 − z)t−1t∑τ=1(t + s − ρ(τ ))α¯f (τ )=∞∑τ=1∞∑t=τ(1 − z)t−1(t + s − ρ(τ ))α¯f (τ )=∞∑τ=1∞∑r=s+1(1 − z)r+τ−s−2rα¯ f (τ )=∞∑τ=1(1 − z)τ−1f (τ )∞∑r=s+1(1 − z)r−1−srα¯= (N gαs )(z)(N f )(z).(12)□Abdeljawad et al. Advances in Difference Equations 2012, 2012:72http://www.advancesindifferenceequations.com/content/2012/1/72Page 3 of 7Lemma 2. Let f be defined on N0. Then(N∇(f (t))(z) = z(N f )(z) − f (0). (13)Proof.(N∇(f (t))(z) =∞∑t=1(1 − z)t−1f (t) −∞∑t=1(1 − z)t−1f (t − 1)=∞∑t=1(1 − z)t−1f (t) −∞∑t=0(1 − z)tf (t)= (N f )(z) − f (0) − (1 − z)∞∑t=1(1 − z)t−1f (t)= (N f )(z) − f (0) − (1 − z)(N f )(z)= z(N f )(z) − f (0).(14)□Lemma 3. Let f be defined on N0and 0 < a ≤ 1. Then(N C∇α0 f )(z) = zα(N f )(z) − zα−1f (0). (15)Proof. By the help of Lemma 1, Proposition 1 and Lemma 2 we have(N C∇α0 f )(z) = (N1(1 − α)g−α0 ∗ ∇f )(z)= zα−1(N∇f )(z)= zα−1[z(N f )(z) − f (0)]= zα(N f )(z) − zα−1f (0).(16)□Lemma 4. Let 0 < a ≤ 1 and f be defined on N0. Then,(i) (NEα¯(λ, t))(z) = zα−1zα−λ.(ii) (NEα,α(λ, t))(z) = λ2zα−λProof. (i) From Example 1 we know thatC∇α0Eα¯(λ, t) = λEα¯(λ, t).Apply the nabla discrete Laplace transform to both sides, then Lemma 3 yields thatzα(NEα¯(λ, t))(z) − zα−1 = λ(NEα¯(λ, t))(z).Hence, it follows.(ii) First it is easy to see that ∇Eα¯(λ, t) = −λ−1Eα,α(λ, t). If we apply N and make useof (i) and Lemma 2, then we have thatNEα,α(λ, t))(z) = λ[z(NEα¯(λ, t))(z) − Eα¯(λ, 0)]= λ[zαzα − λ − 1]=λ2zα − λ .□Abdeljawad et al. Advances in Difference Equations 2012, 2012:72http://www.advancesindifferenceequations.com/content/2012/1/72Page 4 of 73 A discrete Mittag-Leffler Semigroup PropertyBy direct calculations it can be showed that the following semigroup property fordiscrete Mittag-Leffler functionsEα¯(λ, z1)Eα¯(λ, z2) = Eα¯(λ, z1 + z2)does not hold unless l = 0 or a = 1. For more consistency, we next show how cer-tain discrete Mittag-Leffler functions do not satisfy the above mentioned semigroupproperty.Example 2. For 0 < a ≤ 1 and l ≠ 1 we haveEα¯(λ, 1) =∞∑k=0λk1αk(αk + 1)=∞∑k=0λk =11 − λandEα¯(λ, 2) =∞∑k=0λk(αk + 1) =1 + λ(α − 1)(1 − λ)2 .Then, clearlyEα¯(λ, 1)Eα¯(λ, 1) = Eα¯(λ, 1 + 1)Theorem 1. For every real l and 0 < a <1 the following holdsαs∑τ=1t∑r=1(t + s − ρ(τ ) − ρ(r))−α−1Eα¯(λ, τ )Eα¯(λ, r) =t+s∑τ=1(t + s − ρ(τ ))−αEα¯(λ, τ ) −s+1∑τ=1(t + s − ρ(τ ))−αEα¯(λ, τ ) −t∑τ=1(t + s − ρ(τ ))−αEα¯(λ, τ ),(17)for all t, s ∈ N1.Proof. For the sake of simplicity we set f (t) = Eα¯(λ, t). By Definition 4 we have thatfor all t, s ∈ N1,Ct ∇α0 f (t + s) =1(α − 1)t∑τ=1(t − ρ(τ ))−α∇τ f (τ + s)=1(α − 1)t+s∑τ=s+1(t + s − ρ(τ ))α−1∇τ f (τ )= Cr ∇α0 f (r)|r=t+s −1(α − 1)s∑τ=1(t + s − ρ(τ ))−α∇τ f (τ )= λf (t + s) − 1(α − 1)s∑τ=1(t + s − ρ(τ ))−α∇τ f (τ ).(18)In the last equality the fact that Eα¯(λ, t) solves Eq.(8) is used. Applying the nabla dis-crete transform N with respect to t to both sides of (18), we get by Lemma 3 thatzα(N fs)(z) − zα−1f (s) = λ(N fs)(z) − 1(−α)s∑τ=1(N (t + s − ρ(τ ))−α)(z)∇τ f (τ ), (19)where (N fs)(z) and (N (t + s − ρ(τ ))−α)(z) are the nabla discrete transforms of f(t +s) and (t + s − ρ(τ ))−α with respect to t, respectively. By applying the nabla summationAbdeljawad et al. Advances in Difference Equations 2012, 2012:72http://www.advancesindifferenceequations.com/content/2012/1/72Page 5 of 7by parts and making use of (4) we have thats∑τ=1(N (t + s − ρ(τ ))−α)(z)∇τ f (τ ) =f (τ )(N (t + s − τ )−α)(z)|sτ=0 − αs∑τ=1(N (t + s − ρ(τ ))−α−1)(z).(20)By Lemma 1 (i), we arrive ats∑τ=1(N (t + s − ρ(τ ))−α)(z)∇τ f (τ ) = f (s)(1 − α)z1−α− (N (t + s)−α)(z)− αs∑τ=1(N (t + s − ρ(τ ))−α−1)(z),(21)which combining with the equality (19) leads to that(1 − α)zα−1(N fs)(z) =zα−1zα − λ(N (t + s)−α)(z) − αs∑τ=1(N (t + s − ρ(τ ))−α−1)(z) zα−1zα − λ f (τ ).(22)Then applying the inverse nabla discrete Laplace transform to both sides of (22), weget thatt∑τ=1(t − ρ(τ ))−αf (τ + s) =t∑τ=1(t + s − ρ(τ ))−αf (τ ) + αs∑τ=1t∑r=1f (r)(t + s − ρ(τ ) − ρ(r))−α−1f (τ ).(23)Arranging the limit of the sum in the left hand sum of (23) and making use of Pro-position 1 it follows thats+t∑τ=s+1(t − ρ(τ ))−αf (τ ) =t∑τ=1(t + s − ρ(τ ))−αf (τ ) + αs∑τ=1t∑r=1f (r)(t + s − ρ(τ ) − ρ(r))−α−1f (τ ).(24)Splitting the sum in the left hand side of (24) yields directly equality (17).□Remark 2. We note that for a = 1, the summations in (17) are divergent. However, itcan be shown that the semigroup property for E1¯(λ, t) = eˆλ(t, 0) = (1 − λ)−t is just thelimit case state of equality (17) as a ↑ 1. Indeed, if we multiply both sides of (17) with(1 - a) and use summation by parts, then, letting a ↑ 1 we get that the limit state ofthe left is E1¯(λ, t)E1¯(λ, s)and of the right is E1¯(λ, t + s).Author details1Department of Mathematics, Çankaya University, 06530, Ankara, Turkey 2Institute of Space Sciences, P.O.BOX, MG-23,R 76900, Magurele-Bucharest, RomaniaAuthors’ contributionsTA wrote the first draft of the paper, FJ has corrected this draft, DB has prepared the final version of the work and enhancedthe revised version of the manuscript. The authors read carefully and approved the final version of the manuscript.Abdeljawad et al. Advances in Difference Equations 2012, 2012:72http://www.advancesindifferenceequations.com/content/2012/1/72Page 6 of 7Competing interestsThe authors declare that they have no competing interests.Received: 19 March 2012 Accepted: 31 May 2012 Published: 31 May 2012References1. Miller, KS, Ross, B: Fractional difference calculus. Proceedings of the International Symposium on Univalent Functions,Fractional Calculus and Their Applications. pp. 139–152.Nihon University, Koriyama, Japan (1989)2. Podlubny, I: Fractional Differential Equations. Academic Press: San Diego CA (1999)3. Samko, G, Kilbas, AA: Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach,Yverdon (1993)4. Kilbas, A, Srivastava, MH, Trujillo, JJ: Theory and Application of Fractional Differential Equations. pp. 204.North HollandMathematics Studies (2006)5. Magin, RL: Fractional Calculus in Bioengineering. Begell House Publisher, Inc. Connecticut (2006)6. Baleanu, D, Diethelm, K, Scalas, E, Trujillo, JJ: Fractional Calculus Models and Numerical Methods (Series on Complexity,Nonlinearity and Chaos). World Scientific (2012)7. Abdeljawad, T: On Riemann and Caputo fractional differences Original Research Article Computers and Mathematicswith Applications. 62(3), 1602–1611 (2011)8. Abdeljawad, T: Principles of delta and nabla fractional differences (2011). arXiv:1112.5795v19. Atıcı, FM, Eloe, PW: A Transform method in discrete fractional calculus. International Journal of Difference Equations.2(2), 165–176 (2007)10. Atıcı, FM, Eloe, PW: Initial value problems in discrete fractional calculus. Proceedings of the American MathematicalSociety. 137, 981–989 (2009)11. Atıcı, FM, Eloe, WPaul: Discrete fractional calculus with the nabla operator. Electronic Journal of Qualitative Theory ofDifferential equations, Spec Ed I. , 3: 1–12 (2009)12. Atıcı, FM, şengül, S: Modelling with fractional difference equations. 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Birkhäuser, Boston (2003)doi:10.1186/1687-1847-2012-72Cite this article as: Abdeljawad et al.: A semigroup-like Property for Discrete Mittag-Leffler Functions. Advances inDifference Equations 2012 2012:72.Submit your manuscript to a journal and benefi t from:7 Convenient online submission7 Rigorous peer review7 Immediate publication on acceptance7 Open access: articles freely available online7 High visibility within the fi eld7 Retaining the copyright to your article    Submit your next manuscript at 7 springeropen.comAbdeljawad et al. Advances in Difference Equations 2012, 2012:72http://www.advancesindifferenceequations.com/content/2012/1/72Page 7 of 7

A semigroup-like Property for Discrete Mittag-Leffler Functions

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A semigroup-like Property for Discrete Mittag-Leffler Functions

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