# On a singular system of fractional nabla difference equations with boundary conditions

Ioannis K Dassios1* and Dumitru I Baleanu234

Author Affiliations

1 School of Mathematics and Maxwell Institute, The University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom

2 Department of Mathematics and Computer Sciences, Cankaya University, Ankara, Turkey

3 Institute of Space Sciences, Magurele, Bucharest, Romania

4 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia

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Boundary Value Problems 2013, 2013:148  doi:10.1186/1687-2770-2013-148

 Received: 20 February 2013 Accepted: 18 May 2013 Published: 19 June 2013

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.

### Abstract

In this article, we study a boundary value problem of a class of linear singular systems of fractional nabla difference equations whose coefficients are constant matrices. By taking into consideration the cases that the matrices are square with the leading coefficient matrix singular, square with an identically zero matrix pencil and non-square, we provide necessary and sufficient conditions for the existence and uniqueness of solutions. More analytically, we study the conditions under which the boundary value problem has a unique solution, infinite solutions and no solutions. Furthermore, we provide a formula for the case of the unique solution. Finally, numerical examples are given to justify our theory.

##### Keywords:
boundary conditions; singular systems; fractional calculus; nabla operator; difference equations; linear; discrete time system

### 1 Introduction

Difference equations of fractional order have recently proven to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetism, and so forth [1-7]. There has been a significant development in the study of fractional differential/difference equations and inclusions in recent years; see the monographs of Baleanu et al.[1], Kaczorek [4], Klamka et al.[8], Malinowska et al.[5], Podlubny [7], and the survey by Agarwal et al.[9]. For some recent contributions on fractional differential/difference equations, see [1,4,5,8-27] and the references therein. In this article we provide an introductory study for a boundary value problem of a class of singular fractional nabla discrete time systems. If we define by , α integer, and n such that or , then the nabla fractional operator in the case of Riemann-Liouville fractional difference of nth order for any is defined by, see [5,10-12,23],

where the raising power function is defined by

The following problem will then be considered. The singular fractional discrete time systems of the form

(1)

with known boundary conditions

(2)

where (i.e., the algebra of matrices with elements in the field ℱ) with , , , and . For the sake of simplicity, we set and . The matrices F and G can be non-square (when ) or square () and F singular (). The main purpose will be to provide necessary and sufficient conditions for the existence and uniqueness of solutions of the above boundary value problem, i.e., to study the conditions under which the system has unique, infinite and no solutions and to provide a formula for the case of the unique solution (if it exists). Many authors use matrix pencil theory to study linear discrete time systems with constant matrices; see, for instance, [28-43]. A matrix pencil is a family of matrices , parametrized by a complex number s, see [39,41,44,45]. When G is square and , where is the identity matrix, the zeros of the function are the eigenvalues of G. Consequently, the problem of finding the nontrivial solutions of the equation

is called the generalized eigenvalue problem. Although the generalized eigenvalue problem looks like a simple generalization of the usual eigenvalue problem, it exhibits some important differences. In the first place, it is possible for F, G to be non-square matrices. Moreover, even with F, G square it is possible (in the case F is singular) for to be identically zero, independent of s. Finally, even if we assume F, G square matrices with a non-zero pencil, it is possible (when F is singular) for the problem to have infinite eigenvalues. To see this, write the generalized eigenvalue problem in the reciprocal form

If F is singular with a null vector X, then , so that X is an eigenvector of the reciprocal problem corresponding to eigenvalue ; i.e., .

Definition 1.1 Given and an arbitrary , the matrix pencil is called:

1. Regular when and .

2. Singular when or and .

The paper is organized as follows. In Section 2, we study the existence of solutions of the system (1) when its pencil is regular. In Section 3 we study the case of the system (1) with a singular pencil, and Section 3 contains numerical examples.

### 2 Regular case

In this section, we consider the case of the system (1) with a regular pencil. The class of is characterized by a uniquely defined element, known as complex Weierstrass canonical form, , see [39,41,44,45], specified by the complete set of invariants of . This is the set of elementary divisors (e.d.) obtained by factorizing the invariant polynomials into powers of homogeneous polynomials irreducible over the field ℱ. In the case where is regular, we have e.d. of the following type:

• e.d. of the type are called finite elementary divisors (f.e.d.), where is a finite eigenvalue of algebraic multiplicity ;

• e.d. of the type are called infinite elementary divisors (i.e.d.), where q is the algebraic multiplicity of the infinite eigenvalues.

We assume that and .

Definition 2.1 Let be elements of . The direct sum of them denoted by is the .

From the regularity of , there exist nonsingular matrices such that

(3)

and

(4)

The complex Weierstrass form of the regular pencil is defined by

where the first normal Jordan-type element is uniquely defined by the set of the finite eigenvalues of and has the form

The second uniquely defined block corresponds to the infinite eigenvalues of and has the form

The matrix is a nilpotent element of with index , where

and , , are defined as

For algorithms about the computations of the Jordan matrices, see [39,41,44,45].

Definition 2.2 If for the system (1) with boundary conditions (2) there exists at least one solution, the boundary value problem (1)-(2) is said to be consistent.

For the regular matrix pencil of the system (1), there exist nonsingular matrices as applied in (3), (4). Let

(5)

where is a matrix with columns p linear independent (generalized) eigenvectors of the p finite eigenvalues of , and is a matrix with columns q linear independent (generalized) eigenvectors of the q infinite eigenvalues of .

Lemma 2.1Consider the system (1) with a regular pencil. Then the system (1) is divided into two subsystems:

and

Proof Consider the transformation

(6)

and by substituting (6) into (1), we obtain

or, equivalently,

Whereby multiplying by P, we arrive at

Moreover, let

where , , and by using (3) and (4), we obtain

From the above expressions, we arrive easily at the subsystems

(7)

and

(8)

The proof is completed. □

Definition 2.3 With we denote the discrete Mittag-Leffler function with two parameters defined by

(9)

See [10-12,23,46].

Proposition 2.1The subsystem (7) has the solution

(10)

if and only if

(11)

whereis an induced matrix norm andis the discrete Mittag-Leffler function with two parameters as defined by Definition 2.3.

Proof From [10-12,23,46] the solution of (7) can be calculated and given by the formula

or, equivalently, by

The existence and uniqueness of the above solution depends on the convergence of the matrix power series

or, equivalently, if and only if

or, equivalently,

By using the property

we get

or, equivalently,

The proof is completed. □

Proposition 2.2The subsystem (8) has the unique solution

(12)

Proof Let be the index of the nilpotent matrix , i.e., . Then if we obtain the following equations:

by taking the sum of the above equations and using the fact that , we arrive easily at the solution (12). The proof is completed. □

Theorem 2.1Consider the system (1) with a regular pencil and boundary conditions of type (2). Then the boundary value problem (1)-(2) is consistent if and only if:

1. The pencilhaspdistinct eigenvalues and all lie within the open disk

2.

(13)

Furthermore, when the boundary value problem (1)-(2) is consistent, it has a unique solution if and only if:

1.

(14)

2.

(15)

In this case the unique solution is then given by

(16)

whereCis the unique solution of the algebraic system

(17)

Proof By applying the transformation (6) into the system (1), we get the systems (7), (8) with solutions (10), (12) respectively. Note that from Proposition 2.1 the solution (10) exists if and only if

where is the Jordan matrix related to the p finite eigenvalues of the pencil , which is equivalent to the fact that the finite eigenvalues of the pencil must be distinct and all lie within the unit disk . Based on these results, the solution of (1) can be written as

or, equivalently,

or, equivalently, by using (10), (12)

The initial value of the subsystem (7) is not known and can be replaced by a constant vector

The above solution exists if and only if

or, equivalently,

or, equivalently,

For the above algebraic system, there exists at least one solution if and only if

The algebraic system (17) contains equations and p unknowns. Hence the solution is unique if and only if

and

where C is then the unique solution of (17). This can be proved as follows. If we assume that the algebraic system has two solutions and , then

and

or, equivalently,

But the matrix is left invertible since it is assumed to have p linear independent columns and and hence

The unique solution is then given from (16). The proof is completed. □

### 3 Singular case

In this section, we consider the case of the system (1) with a singular pencil. The class of in this case is characterized by a uniquely defined element, , known as the complex Kronecker canonical form, see [39,41,44,45], specified by the complete set of invariants of the singular pencil . This is the set of the elementary divisors (e.d.) and the minimal indices (m.i.). Unlike the case of the regular pencils, where the pencil is characterized only from the e.d., the characterization of a singular matrix pencil apart from the set of the determinantal divisors requires the definition of additional sets of invariants, the minimal indices. The distinguishing feature of a singular pencil is that either or and . Let , be the right and the left null space of a matrix respectively. Then the equations

(18)

(19)

where is the transpose tensor, have solutions in , , which are vectors in the rational vector spaces and respectively. The binary vectors and express dependence relationships among the columns or rows of respectively. Note that and are polynomial vectors. Let and . It is known, see [39,41,44,45], that and as rational vector spaces are spanned by minimal polynomial bases of minimal degrees

(20)

and

(21)

respectively. The set of minimal indices and are known as column minimal indices (c.m.i.) and row minimal indices (r.m.i.) of respectively. To sum up, in the case of a singular pencil, we have invariants of the following type:

finite elementary divisors of the type ;

infinite elementary divisors of the type ;

column minimal indices of the type ;

row minimal indices of the type .

The Kronecker canonical form, see [39,41,44,45], is defined by

(22)

where , are defined as in Section 2. The matrices , , and are defined by

(23)

where for

(24)

where for . The matrices , are defined as

(25)

where for

(26)

where for .

For algorithms about the computations of these matrices, see [39,41,44,45].

Following the above given analysis, there exist nonsingular matrices P, Q with , such that

(27)

Let

(28)

where , , , and .

Lemma 3.1The system (1) is divided into five subsystems:

(29)

the subsystem

(30)

the subsystem

(31)

the subsystem

(32)

and the subsystem

(33)

Proof Consider the transformation

(34)

Substituting the previous expression into (1), we obtain

Whereby multiplying by P and using (27), we arrive at

(35)

Moreover, let

where , , , and . Taking into account the above expressions, we arrive easily at the subsystems (29), (30), (31), (32), and (33). The proof is completed. □

Solving the system (1) is equivalent to solving subsystems (29), (30), (31), (32) and (33). The solutions of the systems (29), (30) are given by (10) and (12) respectively; see Propositions 2.1 and 2.2.

Proposition 3.1The subsystem (31) has infinite solutions and can be taken arbitrarily

(36)

Proof If we set

by using (23), (24), the system (31) can be written as

(37)

Then, for the non-zero blocks, a typical equation from (37) can be written as

(38)

or, equivalently,

or, equivalently,

or, equivalently,

(39)

The system (39) is a regular-type system of difference equations with equations and unknowns. It is clear from the above analysis that in every one of the subsystems one of the coordinates of the solution has to be arbitrary by assigned total. The solution of the system can be assigned arbitrarily

The proof is completed. □

Proposition 3.2The solution of the system (32) is unique and is the zero solution

(40)

Proof From (25), (26) the subsystem (32) can be written as

(41)

Then for the non-zero blocks, a typical equation from (41) can be written as

(42)

or, equivalently,

or, equivalently,

or, equivalently,

(43)

We have a system of +1 difference equations and unknowns. Starting from the last equation, we get the solutions

which means that the solution of the system (32) is unique and is the zero solution. The proof is completed. □

Proposition 3.3The subsystem (33) has an infinite number of solutions that can be taken arbitrarily

(44)

Proof It is easy to observe that the subsystem

does not provide any non-zero equations. Hence all its solutions can be taken arbitrarily. The proof is completed. □

We can now state the following theorem.

Theorem 3.1Consider the system (1) with a singular pencil and known boundary conditions of type (2). Then the boundary value problem (1)-(2) is consistent if and only if:

1.

2. the column minimal indices are zero, i.e.,

(45)

3.

(46)

Furthermore, when the boundary value problem (1)-(2) is consistent, it has a unique solution if and only if

1.

(47)

2.

(48)

In this case the unique solution is given by the formula

(49)

whereCis the unique solution of the algebraic system

(50)

In any other case the system has infinite solutions.

Proof First we consider that the system has non-zero column minimal indices and non-zero row minimal indices. By using the transformation (34), the solutions of the subsystems (29), (30), (31), (32) and (33) are given by (10), (12), (36), (40) and (44) respectively. Note that from Proposition 2.1 the solution (10) exists if and only if

Furthermore, if

Since is unknown, it can be replaced with the unknown vector C. Then

or, equivalently,

Since and can be taken arbitrarily, it is clear that the general singular discrete time system for every suitable defined boundary condition has an infinite number of solutions. It is clear that the existence of the column minimal indices is the reason that the systems (31) and consequently (33) exist. These systems as shown in Propositions 3.1 and 3.3 have always infinite solutions. Thus a necessary condition for the system to have a unique solution is not to have any column minimal indices which are equal to

In this case the Kronecker canonical form of the pencil has the following form:

(51)

Then the system (1) is divided into three subsystems (29), (30), (32) with solutions (10), (12), (40) respectively. Thus

or, equivalently,

The solution exists if and only if

or, equivalently,

or, equivalently,

For the above algebraic system, there exists at least one solution if and only if

The algebraic system (50) contains equations and p unknowns. Hence the solution is unique if and only if

and

where C is then the unique solution of (50). The uniqueness of C can be proved as follows. If we assume that the algebraic system has two solutions and , then

and

or, equivalently,

But the matrix is left invertible since it is assumed to have p linear independent columns and and hence

The unique solution is then given from (49). The proof is completed. □

### 4 Numerical examples

#### Example 1

Assume the system (1) for and . Let

(52)

and

(53)

Then and the pencil is regular. We assume the boundary conditions (2) with

and

The three finite eigenvalues () of the pencil are , 0, , and the Jordan matrix has the form

It is easy to observe that

By calculating the eigenvectors of the finite eigenvalues, we get the matrix

Moreover,

(54)

From (9) we get the Mittag-Leffler function

or, equivalently,

or, equivalently,

or, equivalently,

and since , by using the sum for , we calculate the sum of the matrix power series , and we get

And since

by using the above expression

or, equivalently,

(55)

it is easy to observe that the conditions (13), (14) and (15) are satisfied. Thus from Theorem 2.1 the unique solution of the boundary value problem (1)-(2) is given by

or, equivalently, by

where C is the unique solution of the algebraic system

or, equivalently,

and thus the unique solution of the boundary value problem is

(56)

#### Example 2

We assume the system (1) as in Example 1 but with different boundary conditions. Let

and

It is easy to observe that

since

and thus from Theorem 2.1, and since (13) does not hold, the boundary value problem is not consistent.

#### Example 3

Consider the system (1) and let

Since the matrices F, G are non-square, the matrix pencil is singular and has invariants such as the finite elementary divisors , , an infinite elementary divisor of degree 1 and the row minimal indices , . Since the Jordan matrix has the form

with

for every induced matrix norm, from Theorem 3.1 the boundary value problem (1)-(2) is non-consistent.

#### Example 4

Consider the system (1) for and . Let

(57)

and

(58)

Since the matrices F, G are non-square, the matrix pencil is singular and has invariants such as a finite elementary divisor and the row minimal indices , . We assume the boundary conditions (2) with

and

The Jordan matrix is with for every induced matrix norm. By calculating the matrix , we get

Moreover,

(59)

and since

we get

(60)

By using (59), (60), it is easy to observe that the conditions (45), (46), (47) and (48) are satisfied and thus from Theorem 3.1 the unique solution of the boundary value problem (1)-(2) is given by

or, equivalently, by

where C is the unique solution of the algebraic system

or, equivalently,

and thus the unique solution of the boundary value problem is

(61)

### 5 Conclusions

In this article, we study the boundary value problem of a class of a singular system of fractional nabla difference equations whose coefficients are constant matrices. By taking into consideration the cases that the matrices are square with the leading coefficient singular, square with an identically zero matrix pencil and non-square, we study the conditions under which the boundary value problem has unique, infinite and no solutions. Furthermore, we provide a formula for the case of the unique solution. As a further extension of this article, one can study the stability, the behavior under perturbation and possible applications in economics and engineering of singular matrix difference/differential equations of fractional order. For all this, there is already some research in progress.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

IKD wrote the first draft of the manuscript and DB correct it and prepared the final version of it. All authors read and approved the final manuscript.

### Acknowledgements

We would like to express our sincere gratitude to Professor GI Kalogeropoulos for his helpful and fruitful discussions that clearly improved this article. Moreover, we are very grateful to the anonymous referees for their valuable suggestions that improved the article.

### References

1. Baleanu, D, Diethelm, K, Scalas, E: Fractional Calculus: Models and Numerical Methods, World Scientific, Singapore (2012)

2. Glockle, WG, Nonnenmacher, TF: A fractional calculus approach to self-similar protein dynamics. Biophys. J.. 68(1), 46–53 (1995). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

3. Hilfe R (ed.): Applications of Fractional Calculus in Physics, World Scientific, River Edge (2000)

4. Kaczorek, T: Selected Problems of Fractional Systems Theory, Springer, Berlin (2011)

5. Malinowska, AB, Torres, DFM: Introduction to the Fractional Calculus of Variations, Imperial College Press, London (2012)

6. Metzler, R, Schick, W, Kilian, HG, Nonnenmacher, TF: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys.. 103(16), 7180–7186 (1995). PubMed Abstract | Publisher Full Text

7. Podlubny, I: Fractional Differential Equations, Academic Press, San Diego (1999)

8. Klamka, J: Controllability and minimum energy control problem of fractional discrete-time systems. New Trends in Nanotechnology and Fractional Calculus, pp. 503–509. Springer, New York (2010)

9. Agarwal, RP, Benchohra, M, Hamani, S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math.. 109(3), 973–1033 (2010). Publisher Full Text

10. Ahrendt, K, Castle, L, Holm, M, Yochman, K: Laplace transforms for the nabla-difference operator and a fractional variation of parameters formula. Commun. Appl. Anal. (2011)

11. Atici, FM, Eloe, PW: Linear systems of fractional nabla difference equations. Rocky Mt. J. Math.. 41(2), 353–370 (2011). Publisher Full Text

12. Atici, FM, Eloe, PW: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc.. 137(3), 981–989 (2009)

13. Baleanu, D, Mustafa, OG, Agarwal, RP: Asymptotically linear solutions for some linear fractional differential equations. Abstr. Appl. Anal.. 2010, (2010) Article ID 865139

14. Baleanu, D, Mustafa, OG: On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl.. 59(5), 1835–1841 (2010). Publisher Full Text

15. Baleanu, D, Babakhani, A: Employing of some basic theory for class of fractional differential equations. Adv. Differ. Equ.. 2011, (2011) Article ID 296353

16. Bastos, NRO, Ferreira, RAC, Torres, DFM: Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete Contin. Dyn. Syst.. 29(2), 417–437 (2011)

17. Bastos, NRO, Ferreira, RAC, Torres, DFM: Discrete-time fractional variational problems. Signal Process.. 91(3), 513–524 (2011). Publisher Full Text

18. Bastos, NRO, Mozyrska, D, Torres, DFM: Fractional derivatives and integrals on time scales via the inverse generalized Laplace transform. Int. J. Math. Comput.. 11, 1–9 (2011)

19. Debbouche, A: Fractional nonlocal impulsive quasilinear multi-delay integro-differential systems. Adv. Differ. Equ.. 2011, (2011) Article ID 5

20. Debbouche, A, Baleanu, D: Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput. Math. Appl.. 62(3), 1442–1450 (2011). Publisher Full Text

21. Debbouche, A, Baleanu, D, Agarwal, RP: Nonlocal nonlinear integrodifferential equations of fractional orders. Adv. Differ. Equ.. 2012, (2012) Article ID 78

22. Ferreira, RAC, Torres, DFM: Fractional h-difference equations arising from the calculus of variations. Appl. Anal. Discrete Math.. 5(1), 110–121 (2011). Publisher Full Text

23. Hein, J, McCarthy, Z, Gaswick, N, McKain, B, Speer, K: Laplace transforms for the nabla-difference operator. Panam. Math. J.. 21(3), 79–97 (2011)

24. Kaczorek, T: Positive stable realizations of fractional continuous-time linear systems. Int. J. Appl. Math. Comput. Sci.. 21(4), 697–702 (2011)

25. Kaczorek, T: Application of the Drazin inverse to the analysis of descriptor fractional discrete-time linear systems with regular pencils. Int. J. Appl. Math. Comput. Sci.. 23(1), 29–33 (2013)

26. Klamka, J: Controllability of dynamical systems. Mat. Stosow.. 50(9), 57–75 (2008)

27. Klamka, J, Wyrwał, J: Controllability of second-order infinite-dimensional systems. Syst. Control Lett.. 57(5), 386–391 (2008). Publisher Full Text

28. Dai, L: Singular Control Systems (1988) (edited by M Thoma and A Wyner)

29. Dassios, IK: On non homogeneous linear generalized linear discrete time systems. Circuits Syst. Signal Process.. 31(5), 1699–1712 (2012). Publisher Full Text

30. Dassios, IK, Kalogeropoulos, G: On a non-homogeneous singular linear discrete time system with a singular matrix pencil. Circuits Syst. Signal Process. (2013) doi:10.1007/s00034-012-9541-8

31. Dassios, I: On solutions and algebraic duality of generalized linear discrete time systems. Discrete Math. Appl.. 22(5-6), 665–682 (2012)

32. Dassios, I: On stability and state feedback stabilization of singular linear matrix difference equations. Adv. Differ. Equ.. 2012, (2012) Article ID 75

33. Dassios, I: On robust stability of autonomous singular linear matrix difference equations. Appl. Math. Comput.. 218(12), 6912–6920 (2012). Publisher Full Text

34. Dassios, I: On a boundary value problem of a class of generalized linear discrete time systems. Adv. Differ. Equ.. 2011, (2011) Article ID 51

35. Grispos, E, Giotopoulos, S, Kalogeropoulos, G: On generalised linear discrete-time regular delay systems. J. Inst. Math. Comput. Sci., Math. Ser.. 13(2), 179–187 (2000)

36. Grispos, E, Kalogeropoulos, G, Mitrouli, M: On generalised linear discrete-time singular delay systems. In: Lipitakis EA (ed.) Proceedings of the 5th Hellenic-European Conference on Computer Mathematics and Its Applications (HERCMA 2001), pp. 484–486. LEA, Athens Athens, Greece, September 20-22 2001. (2002)

37. Grispos, E, Kalogeropoulos, G, Stratis, I: On generalised linear discrete-time singular delay systems. J. Math. Anal. Appl.. 245(2), 430–446 (2000). Publisher Full Text

38. Grispos, E: Singular generalised autonomous linear differential systems. Bull. Greek Math. Soc.. 34, 25–43 (1992)

39. Kalogeropoulos, GI: Matrix pencils and linear systems. PhD thesis, City University, London (1985)

40. Kalogeropoulos, G, Stratis, IG: On generalized linear regular delay systems. J. Math. Anal. Appl.. 237(2), 505–514 (1999). Publisher Full Text

41. Karcanias, N, Kalogeropoulos, G: Geometric theory and feedback invariants of generalized linear systems: a matrix pencil approach. Circuits Syst. Signal Process.. 8(3), 375–397 (1989). Publisher Full Text

42. Rugh, WJ: Linear System Theory, Prentice Hall International, London (1996)

43. Sandefur, JT: Discrete Dynamical Systems, Academic Press, San Diego (1990)

44. Gantmacher, RF: The Theory of Matrices. Vols. I, II, Chelsea, New York (1959)

45. Mitrouli, M, Kalogeropoulos, G: A compound matrix algorithm for the computation of the Smith form of a polynomial matrix. Numer. Algorithms. 7(2-4), 145–159 (1994)

46. Nagai, A: Discrete Mittag-Leffler function and its applications. Publ. Res. Inst. Math. Sci.. 1302, 1–20 (2003)