A semigroup-like Property for Discrete Mittag-Leffler Functions
© Abdeljawad et al; licensee Springer. 2012
Received: 19 March 2012
Accepted: 31 May 2012
Published: 31 May 2012
Discrete Mittag-Leffler function of order 0 < α ≤ 1, , λ ≠ 1, satisfies the nabla Caputo fractional linear difference equation
Computations can show that the semigroup identity
does not hold unless λ = 0 or α = 1. In this article we develop a semigroup property for the discrete Mittag-Leffler function in the case α ↑ 1 is just the above identity. The obtained semigroup identity will be useful to develop an operator theory for the discrete fractional Cauchy problem with order α ∈ (0, 1).
KeywordsCaputo fractional difference discrete Mittag-Leffler function discrete nabla Laplace transform convolution
1 Introduction and Preliminaries
The fractional calculus started to be investigated deeply in both theorical and applied view points. One of the main reasons of the fast development of this type of calculus is that it incorporates, as a particular case, the classical calculus. Starting from this interesting point it is natural to ask if there is an extension of the fractional calculus and even more precisely if this generalization could find us new dimensions of some problems within the complex systems which up to now were not solved properly as in Biology, nanotechnology and medicine [1–6]. Mittag-Leffler functions play a very important role in the theory of fractional differential equations [1–4].
Recently there is a huge effort on the line of discretizing the fractional calculus operators and its applications in the control theory and the corresponding variational principles [9–17]. In our opinion the discrete fractional operators can play a crucial role both from the theoretical and applied point of view. However the combinations of the techniques from both fields are not always straithforward namely because the fractional operators are non-local. However, we expect that this new unification will provide 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 and preliminary concepts regarding the rising factorial function, the discrete Mittaf-Leffler functions, Caputo fractional difference and the discrete convolution. In Section 2, we find the nabla discrete transforms of certain discrete Mittag-Leffler functions by making use of the discrete convolution theorem, which will be helpful to proceed in obtaining our main results. Finally, Section 3 deals with a semigroup property for discrete Mittag-Leffler functions and some examples are given to illustrate our results.
For the sake of the nabla fractional calculus we have the following definition
Regarding the rising factorial function we observe the following:
where ρ(t) = t - 1.
If f(t, s) is a function of two variables, we state explicitly to specify to which parameter we apply the transform.
Lemma 1.  For any ,
(i), |1 - z| < 1,
(ii), |1 - z| < b.
where ρ(s) = s - 1 and is the nabla left fractional sum of order α.
Note that if f is defined on , then is defined on .
The first part of the solution is the nabla discrete exponential function . For the sake of more comparisons see reference (, p. 118).
2 Discrete Laplace of Mittag-Leffler Type and Convolution Type Functions
Lemma 4. Let 0 < α ≤ 1 and f be defined on . Then,
- (ii)First it is easy to see that . If we apply and make use of (i) and Lemma 2, then we have that
3 A discrete Mittag-Leffler Semigroup Property
does not hold unless λ = 0 or α = 1. For more consistency, we next show how certain discrete Mittag-Leffler functions do not satisfy the above mentioned semigroup property.
for all t, .
Splitting the sum in the left hand side of (24) yields directly equality (17).
Remark 2. We note that for α = 1, the summations in (17) are divergent. However, it can be shown that the semigroup property for is just the limit case state of equality (17) as α↑ 1. Indeed, if we multiply both sides of (17) with (1 - α) and use summation by parts, then, letting α ↑ 1 we get that the limit state of the left is and of the right is .
- Miller KS, Ross B: Fractional difference calculus. In Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications. Nihon University, Koriyama, Japan; 1989:139–152.Google Scholar
- Podlubny I: Fractional Differential Equations. Academic Press: San Diego CA; 1999.MATHGoogle Scholar
- Samko G, Kilbas AA: Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon; 1993.MATHGoogle Scholar
- Kilbas A, Srivastava MH, Trujillo JJ: Theory and Application of Fractional Differential Equations. North Holland Mathematics Studies; 2006:204.Google Scholar
- Magin RL: Fractional Calculus in Bioengineering. Begell House Publisher, Inc. Connecticut; 2006.Google Scholar
- Baleanu D, Diethelm K, Scalas E, Trujillo JJ: Fractional Calculus Models and Numerical Methods (Series on Complexity, Nonlinearity and Chaos). World Scientific; 2012.MATHGoogle Scholar
- Abdeljawad T: On Riemann and Caputo fractional differences Original Research Article Computers and Mathematics with Applications. 2011, 62(3):1602–1611.Google Scholar
- Abdeljawad T: Principles of delta and nabla fractional differences. 2011. arXiv:1112.5795v1Google Scholar
- Atıcı FM, Eloe PW: A Transform method in discrete fractional calculus. International Journal of Difference Equations 2007, 2(2):165–176.MathSciNetGoogle Scholar
- Atıcı FM, Eloe PW: Initial value problems in discrete fractional calculus. Proceedings of the American Mathematical Society 2009, 137: 981–989.MathSciNetMATHGoogle Scholar
- Atıcı FM, Eloe PaulW: Discrete fractional calculus with the nabla operator. Electronic Journal of Qualitative Theory of Differential equations, Spec Ed I 2009, (3):1–12.
- Atıcı FM, şengül S: Modelling with fractional difference equations. Journal of Mathematical Analysis and Applications 2010, 369: 1–9. 10.1016/j.jmaa.2010.02.009MathSciNetView ArticleMATHGoogle Scholar
- Bastos NunoRO, Ferreira RuiAC, Torres DelfimFM: Discrete-time fractional variational problems. Signal Processing 2011, 91(3):513–524. 10.1016/j.sigpro.2010.05.001View ArticleMATHGoogle Scholar
- Gray HL, Zhang NF: On a new definition of the fractional difference. Math-ematicsof Computaion 1988, 50(182):513–529.MathSciNetView ArticleMATHGoogle Scholar
- Anastassiou GA: Principles of delta fractional calculus on time scales and inequalities. Mathematical and Computer Modelling 2010, 52: 556–566. 10.1016/j.mcm.2010.03.055MathSciNetView ArticleMATHGoogle Scholar
- Anastassiou GA: Foundations of nabla fractional calculus on time scales and inequalities. Computer and Mathematics with Applications 2010, 59: 3750–3762. 10.1016/j.camwa.2010.03.072MathSciNetView ArticleMATHGoogle Scholar
- Abdeljawad T, Baleanu D: Fractional Differences and integration by parts. Journal of Computational Analysis and Applications 2011, 13(3):574–582.MathSciNetMATHGoogle Scholar
- Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston; 2003.View ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.