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Abstract and Applied Analysis
Volumeย 2011ย (2011), Article IDย 565067, 21 pages
http://dx.doi.org/10.1155/2011/565067
Research Article

Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations

Institute of Mathematics, Faculty of Mechanical Engineering, Technickรก 2, 616 69 Brno, Czech Republic

Received 7 January 2011; Accepted 23 April 2011

Academic Editor: Yuri V.ย Rogovchenko

Copyright ยฉ 2011 Jan ฤŒermรกk et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates some initial value problems in discrete fractional calculus. We introduce a linear difference equation of fractional order along with suitable initial conditions of fractional type and prove the existence and uniqueness of the solution. Then the structure of the solutions space is discussed, and, in a particular case, an explicit form of the general solution involving discrete analogues of Mittag-Leffler functions is presented. All our observations are performed on a special time scale which unifies and generalizes ordinary difference calculus and ๐‘ž-difference calculus. Some of our results are new also in these particular discrete settings.

1. Introduction

The fractional calculus is a research field of mathematical analysis which may be taken for an old as well as a modern topic. It is an old topic because of its long history starting from some notes and ideas of G. W. Leibniz and L. Euler. On the other hand, it is a modern topic due to its enormous development during the last two decades. The present interest of many scientists and engineers in the theory of fractional calculus has been initiated by applications of this theory as well as by new mathematical challenges.

The theory of discrete fractional calculus belongs among these challenges. Foundations of this theory were formulated in pioneering works by Agarwal [1] and Diaz and Osler [2], where basic approaches, definitions, and properties of the theory of fractional sums and differences were reported (see also [3, 4]). The cited papers discussed these notions on discrete sets formed by arithmetic or geometric sequences (giving rise to fractional difference calculus or ๐‘ž-difference calculus). Recently, a series of papers continuing this research has appeared (see, e.g., [5, 6]).

The extension of basic notions of fractional calculus to other discrete settings was performed in [7], where fractional sums and differences have been introduced and studied in the framework of (๐‘ž,โ„Ž)-calculus, which can be reduced to ordinary difference calculus and ๐‘ž-difference calculus via the choice ๐‘ž=โ„Ž=1 and โ„Ž=0, respectively. This extension follows recent trends in continuous and discrete analysis, characterized by a unification and generalization, and resulting into the origin and progressive development of the time scales theory (see [8, 9]). Discussing problems of fractional calculus, a question concerning the introduction of (Hilger) fractional derivative or integral on arbitrary time scale turns out to be a difficult matter. Although first attempts have been already performed (see, e.g., [10]), results obtained in this direction seem to be unsatisfactory.

The aim of this paper is to introduce some linear nabla (๐‘ž,โ„Ž)-fractional difference equations (i.e., equations involving difference operators of noninteger orders) and investigate their basic properties. Some particular results concerning this topic are already known, either for ordinary difference equations or ๐‘ž-difference equations of fractional order (some relevant references will be mentioned in Section 4). We wish to unify them and also present results which are new even also in these particular discrete settings.

The structure of the paper is the following: Section 2 presents a necessary mathematical background related to discrete fractional calculus. In particular, we are going to make some general remarks concerning fractional calculus on arbitrary time scales. In Section 3, we consider a linear nabla (๐‘ž,โ„Ž)-difference equation of noninteger order and discuss the question of the existence and uniqueness of the solution for the corresponding initial value problem, as well as the question of a general solution of this equation. In Section 4, we consider a particular case of the studied equation and describe the base of its solutions space by the use of eigenfunctions of the corresponding difference operator. We show that these eigenfunctions can be taken for discrete analogues of the Mittag-Leffler functions.

2. Preliminaries

The basic definitions of fractional calculus on continuous or discrete settings usually originate from the Cauchy formula for repeated integration or summation, respectively. We state here its general form valid for arbitrary time scale ๐•‹. Before doing this, we recall the notion of Taylor monomials introduced in [9]. These monomials ๎โ„Ž๐‘›โˆถ๐•‹2โ†’โ„, ๐‘›โˆˆโ„•0 are defined recursively as follows: ๎โ„Ž0(๐‘ก,๐‘ )=1โˆ€๐‘ ,๐‘กโˆˆ๐•‹(2.1) and, given ๎โ„Ž๐‘› for ๐‘›โˆˆโ„•0, we have ๎โ„Ž๐‘›+1(๎€œ๐‘ก,๐‘ )=๐‘ก๐‘ ๎โ„Ž๐‘›(๐œ,๐‘ )โˆ‡๐œโˆ€๐‘ ,๐‘กโˆˆ๐•‹.(2.2) Now let ๐‘“โˆถ๐•‹โ†’โ„ be โˆ‡-integrable on [๐‘Ž,๐‘]โˆฉ๐•‹, ๐‘Ž,๐‘โˆˆ๐•‹. We put ๐‘Žโˆ‡โˆ’1๎€œ๐‘“(๐‘ก)=๐‘ก๐‘Ž๐‘“(๐œ)โˆ‡๐œโˆ€๐‘กโˆˆ๐•‹,๐‘Žโ‰ค๐‘กโ‰ค๐‘(2.3) and define recursively ๐‘Žโˆ‡โˆ’๐‘›๎€œ๐‘“(๐‘ก)=๐‘ก๐‘Ž๐‘Žโˆ‡โˆ’๐‘›+1๐‘“(๐œ)โˆ‡๐œ(2.4) for ๐‘›=2,3,โ€ฆ. Then we have the following.

Proposition 2.1 (Nabla Cauchy formula). Let ๐‘›โˆˆโ„ค+, ๐‘Ž,๐‘โˆˆ๐•‹ and let ๐‘“โˆถ๐•‹โ†’โ„ be โˆ‡-integrable on [๐‘Ž,๐‘]โˆฉ๐•‹. If ๐‘กโˆˆ๐•‹, ๐‘Žโ‰ค๐‘กโ‰ค๐‘, then ๐‘Žโˆ‡โˆ’๐‘›๎€œ๐‘“(๐‘ก)=๐‘ก๐‘Ž๎โ„Ž๐‘›โˆ’1(๐‘ก,๐œŒ(๐œ))๐‘“(๐œ)โˆ‡๐œ.(2.5)

Proof. This assertion can be proved by induction. If ๐‘›=1, then (2.5) obviously holds. Let ๐‘›โ‰ฅ2 and assume that (2.5) holds with ๐‘› replaced with ๐‘›โˆ’1, that is, ๐‘Žโˆ‡โˆ’๐‘›+1๎€œ๐‘“(๐‘ก)=๐‘ก๐‘Ž๎โ„Ž๐‘›โˆ’2(๐‘ก,๐œŒ(๐œ))๐‘“(๐œ)โˆ‡๐œ.(2.6) By the definition, the left-hand side of (2.5) is an antiderivative of ๐‘Žโˆ‡โˆ’๐‘›+1๐‘“(๐‘ก). We show that the right-hand side of (2.5) is an antiderivative of โˆซ๐‘ก๐‘Ž๎โ„Ž๐‘›โˆ’2(๐‘ก,๐œŒ(๐œ))๐‘“(๐œ)โˆ‡๐œ. Indeed, it holds โˆ‡๎€œ๐‘ก๐‘Ž๎โ„Ž๐‘›โˆ’1(๎€œ๐‘ก,๐œŒ(๐œ))๐‘“(๐œ)โˆ‡๐œ=๐‘ก๐‘Žโˆ‡๎โ„Ž๐‘›โˆ’1(๎€œ๐‘ก,๐œŒ(๐œ))๐‘“(๐œ)โˆ‡๐œ=๐‘ก๐‘Ž๎โ„Ž๐‘›โˆ’2(๐‘ก,๐œŒ(๐œ))๐‘“(๐œ)โˆ‡๐œ,(2.7) where we have employed the property โˆ‡๎€œ๐‘ก๐‘Ž๎€œ๐‘”(๐‘ก,๐œ)โˆ‡๐œ=๐‘ก๐‘Žโˆ‡๐‘”(๐‘ก,๐œ)โˆ‡๐œ+๐‘”(๐œŒ(๐‘ก),๐‘ก)(2.8) (see [9, page 139]). Consequently, the relation (2.5) holds up to a possible additive constant. Substituting ๐‘ก=๐‘Ž, we can find this additive constant zero.

The formula (2.5) is a corner stone in the introduction of the nabla fractional integral ๐‘Žโˆ‡โˆ’๐›ผ๐‘“(๐‘ก) for positive reals ๐›ผ. However, it requires a reasonable and natural extension of a discrete system of monomials (๎โ„Ž๐‘›,๐‘›โˆˆโ„•0) to a continuous system (๎โ„Ž๐›ผ,๐›ผโˆˆโ„+). This matter is closely related to a problem of an explicit form of ๎โ„Ž๐‘›. Of course, it holds ๎โ„Ž1(๐‘ก,๐‘ )=๐‘กโˆ’๐‘  for all ๐‘ก,๐‘ โˆˆ๐•‹. However, the calculation of ๎โ„Ž๐‘› for ๐‘›>1 is a difficult task which seems to be answerable only in some particular cases. It is well known that for ๐•‹=โ„, it holds ๎โ„Ž๐‘›(๐‘ก,๐‘ )=(๐‘กโˆ’๐‘ )๐‘›๐‘›!,(2.9) while for discrete time scales ๐•‹=โ„ค and ๐•‹=๐‘žโ„ค={๐‘ž๐‘˜,๐‘˜โˆˆโ„ค}โˆช{0}, ๐‘ž>1, we have ๎โ„Ž๐‘›โˆ(๐‘ก,๐‘ )=๐‘›โˆ’1๐‘—=0(๐‘กโˆ’๐‘ +๐‘—),๎โ„Ž๐‘›!๐‘›(๐‘ก,๐‘ )=๐‘›โˆ’1๎‘๐‘—=0๐‘ž๐‘—๐‘กโˆ’๐‘ โˆ‘๐‘—๐‘Ÿ=0๐‘ž๐‘Ÿ,(2.10) respectively. In this connection, we recall a conventional notation used in ordinary difference calculus and ๐‘ž-calculus, namely,(๐‘กโˆ’๐‘ )(๐‘›)=๐‘›โˆ’1๎‘๐‘—=0(๐‘กโˆ’๐‘ +๐‘—),(๐‘กโˆ’๐‘ )(๐‘›)ฬƒ๐‘ž=๐‘ก๐‘›๐‘›โˆ’1๎‘๐‘—=0๎‚ต1โˆ’ฬƒ๐‘ž๐‘—๐‘ ๐‘ก๎‚ถ(0<ฬƒ๐‘ž<1)(2.11) and [๐‘—]๐‘ž=โˆ‘๐‘—โˆ’1๐‘Ÿ=0๐‘ž๐‘Ÿ(๐‘ž>0), [๐‘›]๐‘žโˆ!=๐‘›๐‘—=1[๐‘—]๐‘ž. To extend the meaning of these symbols also for noninteger values (as it is required in the discrete fractional calculus), we recall some other necessary background of ๐‘ž-calculus. For any ๐‘ฅโˆˆโ„ and 0<๐‘žโ‰ 1, we set [๐‘ฅ]๐‘ž=(๐‘ž๐‘ฅโˆ’1)/(๐‘žโˆ’1). By the continuity, we put [๐‘ฅ]1=๐‘ฅ. Further, the ๐‘ž-Gamma function is defined for 0<ฬƒ๐‘ž<1 as ฮ“ฬƒ๐‘ž(๐‘ฅ)=(ฬƒ๐‘ž,ฬƒ๐‘ž)โˆž(1โˆ’ฬƒ๐‘ž)1โˆ’๐‘ฅ(ฬƒ๐‘ž๐‘ฅ,ฬƒ๐‘ž)โˆž,(2.12) where (๐‘,ฬƒ๐‘ž)โˆž=โˆโˆž๐‘—=0(1โˆ’๐‘ฬƒ๐‘ž๐‘—), ๐‘ฅโˆˆโ„โงต{0,โˆ’1,โˆ’2,โ€ฆ}. Note that this function satisfies the functional relation ฮ“ฬƒ๐‘ž(๐‘ฅ+1)=[๐‘ฅ]ฬƒ๐‘žฮ“ฬƒ๐‘ž(๐‘ฅ) and the condition ฮ“ฬƒ๐‘ž(1)=1. Using this, the ๐‘ž-binomial coefficient can be introduced as ๎‚ธ๐‘ฅ๐‘˜๎‚นฬƒ๐‘ž=ฮ“ฬƒ๐‘ž(๐‘ฅ+1)ฮ“ฬƒ๐‘ž(๐‘˜+1)ฮ“ฬƒ๐‘ž(๐‘ฅโˆ’๐‘˜+1),๐‘ฅโˆˆโ„,๐‘˜โˆˆโ„ค.(2.13) Note that although the ๐‘ž-Gamma function is not defined at nonpositive integers, the formula ฮ“ฬƒ๐‘ž(๐‘ฅ+๐‘š)ฮ“ฬƒ๐‘ž(๐‘ฅ)=(โˆ’1)๐‘šฬƒ๐‘ž๎€ท๐‘š2๎€ธ๐‘ฅ๐‘š+ฮ“ฬƒ๐‘ž(1โˆ’๐‘ฅ)ฮ“ฬƒ๐‘ž(1โˆ’๐‘ฅโˆ’๐‘š),๐‘ฅโˆˆโ„,๐‘šโˆˆโ„ค+(2.14) permits to calculate this ratio also at such the points. It is well known that if ฬƒ๐‘žโ†’1โˆ’ then ฮ“ฬƒ๐‘ž(๐‘ฅ) becomes the Euler Gamma function ฮ“(๐‘ฅ) (and analogously for the ๐‘ž-binomial coefficient). Among many interesting properties of the ๐‘ž-Gamma function and ๐‘ž-binomial coefficients, we mention ๐‘ž-Pascal rules๎‚ธ๐‘ฅ๐‘˜๎‚นฬƒ๐‘ž=๎‚ธ๎‚น๐‘ฅโˆ’1๐‘˜โˆ’1ฬƒ๐‘ž+ฬƒ๐‘ž๐‘˜๎‚ธ๐‘˜๎‚น๐‘ฅโˆ’1ฬƒ๐‘ž๎‚ธ๐‘ฅ๐‘˜๎‚น,๐‘ฅโˆˆโ„,๐‘˜โˆˆโ„ค,(2.15)ฬƒ๐‘ž=ฬƒ๐‘ž๐‘ฅโˆ’๐‘˜๎‚ธ๎‚น๐‘ฅโˆ’1๐‘˜โˆ’1ฬƒ๐‘ž+๎‚ธ๐‘˜๎‚น๐‘ฅโˆ’1ฬƒ๐‘ž,๐‘ฅโˆˆโ„,๐‘˜โˆˆโ„ค(2.16) and the ๐‘ž-Vandermonde identity ๐‘š๎“๐‘—=0๎‚ธ๐‘ฅ๎‚น๐‘šโˆ’๐‘—ฬƒ๐‘ž๎‚ธ๐‘ฆ๐‘—๎‚นฬƒ๐‘žฬƒ๐‘ž๐‘—2โˆ’๐‘š๐‘—+๐‘ฅ๐‘—=๎‚ธ๐‘š๎‚น๐‘ฅ+๐‘ฆฬƒ๐‘ž,๐‘ฅ,๐‘ฆโˆˆโ„,๐‘šโˆˆโ„•0(2.17) (see [11]) that turn out to be very useful in our further investigations.

The computation of an explicit form of ๎โ„Ž๐‘›(๐‘ก,๐‘ ) can be performed also in a more general case. We consider here the time scale ๐•‹๐‘ก0(๐‘ž,โ„Ž)=๎€ฝ๐‘ก0๐‘ž๐‘˜+[๐‘˜]๐‘ž๎€พโˆช๎‚ปโ„Žโ„Ž,๐‘˜โˆˆโ„ค๎‚ผ1โˆ’๐‘ž,๐‘ก0>0,๐‘žโ‰ฅ1,โ„Žโ‰ฅ0,๐‘ž+โ„Ž>1(2.18) (see also [7]). Note that if ๐‘ž=1 then the cluster point โ„Ž/(1โˆ’๐‘ž)=โˆ’โˆž is not involved in ๐•‹๐‘ก0(๐‘ž,โ„Ž). The forward and backward jump operator is the linear function ๐œŽ(๐‘ก)=๐‘ž๐‘ก+โ„Ž and ๐œŒ(๐‘ก)=๐‘žโˆ’1(๐‘กโˆ’โ„Ž), respectively. Similarly, the forward and backward graininess is given by ๐œ‡(๐‘ก)=(๐‘žโˆ’1)๐‘ก+โ„Ž and ๐œˆ(๐‘ก)=๐‘žโˆ’1๐œ‡(๐‘ก), respectively. In particular, if ๐‘ก0=๐‘ž=โ„Ž=1, then ๐•‹๐‘ก0(๐‘ž,โ„Ž) becomes โ„ค, and if ๐‘ก0=1, ๐‘ž>1, โ„Ž=0, then ๐•‹๐‘ก0(๐‘ž,โ„Ž) is reduced to ๐‘žโ„ค.

Let ๐‘Žโˆˆ๐•‹๐‘ก0(๐‘ž,โ„Ž), ๐‘Ž>โ„Ž/(1โˆ’๐‘ž) be fixed. Then we introduce restrictions of the time scale ๐•‹๐‘ก0(๐‘ž,โ„Ž) by the relation ๎‚๐•‹๐œŽ๐‘–(๐‘Ž)(๐‘ž,โ„Ž)=๎‚†๐‘กโˆˆ๐•‹๐‘ก0(๐‘ž,โ„Ž),๐‘กโ‰ฅ๐œŽ๐‘–๎‚‡(๐‘Ž),๐‘–=0,1,โ€ฆ,(2.19) where the symbol ๐œŽ๐‘– stands for the ๐‘–th iterate of ๐œŽ (analogously, we use the symbol ๐œŒ๐‘–). To simplify the notation, we put ฬƒ๐‘ž=1/๐‘ž whenever considering the time scale ๐•‹๐‘ก0(๐‘ž,โ„Ž) or ๎‚๐•‹๐œŽ๐‘–(๐‘Ž)(๐‘ž,โ„Ž).

Using the induction principle, we can verify that Taylor monomials on ๐•‹๐‘ก0(๐‘ž,โ„Ž) have the form ๎โ„Ž๐‘›(โˆ๐‘ก,๐‘ )=๐‘›โˆ’1๐‘—=0๎€ท๐œŽ๐‘—๎€ธ(๐‘ก)โˆ’๐‘ [๐‘›]๐‘ž!=โˆ๐‘›โˆ’1๐‘—=0๎€ท๐‘กโˆ’๐œŒ๐‘—๎€ธ(๐‘ )[๐‘›]ฬƒ๐‘ž!.(2.20) Note that this result generalizes previous forms (2.10) and, moreover, enables its unified notation. In particular, if we introduce the symbolic (๐‘ž,โ„Ž)-power(๐‘กโˆ’๐‘ )((๐‘›)ฬƒ๐‘ž,โ„Ž)=๐‘›โˆ’1๎‘๐‘—=0๎€ท๐‘กโˆ’๐œŒ๐‘—๎€ธ(๐‘ )(2.21) unifying (2.11), then the Cauchy formula (2.5) can be rewritten for ๐•‹=๐•‹๐‘ก0(๐‘ž,โ„Ž) as ๐‘Žโˆ‡โˆ’๐‘›๎€œ๐‘“(๐‘ก)=๐‘ก๐‘Ž(๐‘กโˆ’๐œŒ(๐œ))๎€ท(๐‘›โˆ’1)ฬƒ๐‘ž๎€ธ,โ„Ž[]๐‘›โˆ’1ฬƒ๐‘ž!๐‘“(๐œ)โˆ‡๐œ.(2.22)

Discussing a reasonable generalization of (๐‘ž,โ„Ž)-power (2.21) to real values ๐›ผ instead of integers ๐‘›, we recall broadly accepted extensions of its particular cases (2.11) in the form (๐‘กโˆ’๐‘ )(๐›ผ)=ฮ“(๐‘กโˆ’๐‘ +๐›ผ)ฮ“(๐‘กโˆ’๐‘ ),(๐‘กโˆ’๐‘ )(๐›ผ)ฬƒ๐‘ž=๐‘ก๐›ผ(๐‘ /๐‘ก,ฬƒ๐‘ž)โˆž(ฬƒ๐‘ž๐›ผ๐‘ /๐‘ก,ฬƒ๐‘ž)โˆž,๐‘กโ‰ 0.(2.23) Now, we assume ๐‘ ,๐‘กโˆˆ๐•‹๐‘ก0(๐‘ž,โ„Ž), ๐‘กโ‰ฅ๐‘ >โ„Ž/(1โˆ’๐‘ž). First, consider (๐‘ž,โ„Ž)-power (2.21) corresponding to the time scale ๐•‹๐‘ก0(๐‘ž,โ„Ž), where ๐‘ž>1. Then we can rewrite (2.21) as (๐‘กโˆ’๐‘ )๎€ท(๐‘›)ฬƒ๐‘ž๎€ธ,โ„Ž=๎‚ต๐‘ก+โ„Žฬƒ๐‘ž๎‚ถ1โˆ’ฬƒ๐‘ž๐‘›๐‘›โˆ’1๎‘๐‘—=0๎‚ต1โˆ’ฬƒ๐‘ž๐‘—๐‘ +โ„Žฬƒ๐‘ž/(1โˆ’ฬƒ๐‘ž)๎‚ถ[๐‘ก]โˆ’[๐‘ ])๐‘ก+โ„Žฬƒ๐‘ž/(1โˆ’ฬƒ๐‘ž)=((๐‘›)ฬƒ๐‘ž,(2.24) where [๐‘ก]=๐‘ก+โ„Žฬƒ๐‘ž/(1โˆ’ฬƒ๐‘ž) and [๐‘ ]=๐‘ +โ„Žฬƒ๐‘ž/(1โˆ’ฬƒ๐‘ž). A required extension of (๐‘ž,โ„Ž)-power (2.21) is then provided by the formula (๐‘กโˆ’๐‘ )((๐›ผ)ฬƒ๐‘ž,โ„Ž)[๐‘ก]โˆ’[๐‘ ])=((๐›ผ)ฬƒ๐‘ž.(2.25) Now consider (๐‘ž,โ„Ž)-power (2.21) corresponding to the time scale ๐•‹๐‘ก0(๐‘ž,โ„Ž), where ๐‘ž=1. Then (๐‘กโˆ’๐‘ )(๐‘›)(1,โ„Ž)=๐‘›โˆ’1๎‘๐‘—=0(๐‘กโˆ’๐‘ +๐‘—โ„Ž)=โ„Ž๐‘›๐‘›โˆ’1๎‘๐‘—=0๎‚€๐‘กโˆ’๐‘ โ„Ž๎‚+๐‘—=โ„Ž๐‘›((๐‘กโˆ’๐‘ )/โ„Ž+๐‘›โˆ’1)!((๐‘กโˆ’๐‘ )/โ„Žโˆ’1)!(2.26) and the formula (2.21) can be extended by (๐‘กโˆ’๐‘ )(๐›ผ)(1,โ„Ž)=โ„Ž๐›ผฮ“((๐‘กโˆ’๐‘ )/โ„Ž+๐›ผ)ฮ“((๐‘กโˆ’๐‘ )/โ„Ž).(2.27) These definitions are consistent, since it can be shown that limฬƒ๐‘žโ†’1โˆ’([๐‘ก]โˆ’[๐‘ ])(๐›ผ)ฬƒ๐‘ž=(๐‘กโˆ’๐‘ )(๐›ผ)(1,โ„Ž).(2.28) Now the required extension of the monomial ๎โ„Ž๐‘›(๐‘ก,๐‘ ) corresponding to ๐•‹๐‘ก0(๐‘ž,โ„Ž) takes the form ๎โ„Ž๐›ผ(๐‘ก,๐‘ )=(๐‘กโˆ’๐‘ )((๐›ผ)ฬƒ๐‘ž,โ„Ž)ฮ“ฬƒ๐‘ž(๐›ผ+1).(2.29)

Another (equivalent) expression of ๎โ„Ž๐›ผ(๐‘ก,๐‘ ) is provided by the following assertion.

Proposition 2.2. Let ๐›ผโˆˆโ„, ๐‘ ,๐‘กโˆˆ๐•‹๐‘ก0(๐‘ž,โ„Ž) and ๐‘›โˆˆโ„•0 be such that ๐‘ก=๐œŽ๐‘›(๐‘ ). Then ๎โ„Ž๐›ผ(๐‘ก,๐‘ )=(๐œˆ(๐‘ก))๐›ผ๎‚ธ๎‚น๐›ผ+๐‘›โˆ’1๐‘›โˆ’1ฬƒ๐‘ž=(๐œˆ(๐‘ก))๐›ผ๎‚ธ๎‚นโˆ’๐›ผโˆ’1๐‘›โˆ’1ฬƒ๐‘ž(โˆ’1)๐‘›โˆ’1ฬƒ๐‘ž๎€ท๐‘›2๎€ธ๐›ผ(๐‘›โˆ’1)+.(2.30)

Proof. Let ๐‘ž>1. Using the relations [๐‘ก]=๐œˆ(๐‘ก)(,[๐‘ ]1โˆ’ฬƒ๐‘ž)[๐‘ก]=ฬƒ๐‘ž๐‘›,(2.31) we can derive that ๎โ„Ž๐›ผ[๐‘ก](๐‘ก,๐‘ )=๐›ผ([๐‘ ]/[๐‘ก]),ฬƒ๐‘žโˆžฮ“ฬƒ๐‘ž(๐›ผ+1)(ฬƒ๐‘ž๐›ผ[๐‘ ]/[๐‘ก],ฬƒ๐‘ž)โˆž=(1โˆ’ฬƒ๐‘ž)โˆ’๐›ผ๐œˆ(๐‘ก)๐›ผ(ฬƒ๐‘ž๐‘›),ฬƒ๐‘žโˆžฮ“ฬƒ๐‘ž๎€ท(๐›ผ+1)ฬƒ๐‘ž๐›ผ+๐‘›๎€ธ,ฬƒ๐‘žโˆž=(๐œˆ(๐‘ก))๐›ผฮ“ฬƒ๐‘ž(๐›ผ+๐‘›)ฮ“ฬƒ๐‘ž(๐›ผ+1)ฮ“ฬƒ๐‘ž(๐‘›)=(๐œˆ(๐‘ก))๐›ผ๎‚ธ๎‚น๐›ผ+๐‘›โˆ’1๐‘›โˆ’1ฬƒ๐‘ž.(2.32) The second equality in (2.30) follows from the identity (2.14). The case ๐‘ž=1 results from (2.27).

The key property of ๎โ„Ž๐›ผ(๐‘ก,๐‘ ) follows from its differentiation. The symbol โˆ‡๐‘š(๐‘ž,โ„Ž) used in the following assertion (and also undermentioned) is the ๐‘šth order nabla (๐‘ž,โ„Ž)-derivative on the time scale ๐•‹๐‘ก0(๐‘ž,โ„Ž), defined for ๐‘š=1 as โˆ‡(๐‘ž,โ„Ž)๐‘“(๐‘ก)=๐‘“(๐‘ก)โˆ’๐‘“(๐œŒ(๐‘ก))=๐œˆ(๐‘ก)๐‘“(๐‘ก)โˆ’๐‘“(ฬƒ๐‘ž(๐‘กโˆ’โ„Ž))(1โˆ’ฬƒ๐‘ž)๐‘ก+ฬƒ๐‘žโ„Ž(2.33) and iteratively for higher orders.

Lemma 2.3. Let ๐‘šโˆˆโ„ค+, ๐›ผโˆˆโ„, ๐‘ ,๐‘กโˆˆ๐•‹๐‘ก0(๐‘ž,โ„Ž) and ๐‘›โˆˆโ„ค+, ๐‘›โ‰ฅ๐‘š be such that ๐‘ก=๐œŽ๐‘›(๐‘ ). Then โˆ‡๐‘š(๐‘ž,โ„Ž)๎โ„Ž๐›ผ๎ƒฏ๎โ„Ž(๐‘ก,๐‘ )=๐›ผโˆ’๐‘š(๐‘ก,๐‘ ),๐›ผโˆ‰{0,1,โ€ฆ,๐‘šโˆ’1},0,๐›ผโˆˆ{0,1,โ€ฆ,๐‘šโˆ’1}.(2.34)

Proof. First let ๐‘š=1. For ๐›ผ=0 we get ๎โ„Ž0(๐‘ก,๐‘ )=1 and the first nabla (๐‘ž,โ„Ž)-derivative is zero. If ๐›ผโ‰ 0, then by (2.30) and (2.16), we have โˆ‡(๐‘ž,โ„Ž)๎โ„Ž๐›ผ(๎โ„Ž๐‘ก,๐‘ )=๐›ผ๎โ„Ž(๐‘ก,๐‘ )โˆ’๐›ผ(๐œŒ(๐‘ก),๐‘ )=1๐œˆ(๐‘ก)๎ƒฉ๐œˆ(๐‘ก)(๐œˆ(๐‘ก))๐›ผ๎‚ธ๎‚น๐›ผ+๐‘›โˆ’1๐‘›โˆ’1ฬƒ๐‘žโˆ’(๐œˆ(๐œŒ(๐‘ก)))๐›ผ๎‚ธ๎‚น๐›ผ+๐‘›โˆ’2๐‘›โˆ’2ฬƒ๐‘ž๎ƒช=(๐œˆ(๐‘ก))๐›ผโˆ’1๎ƒฉ๎‚ธ๎‚น๐›ผ+๐‘›โˆ’1๐‘›โˆ’1ฬƒ๐‘žโˆ’ฬƒ๐‘ž๐›ผ๎‚ธ๎‚น๐›ผ+๐‘›โˆ’2๐‘›โˆ’2ฬƒ๐‘ž๎ƒช=๎โ„Ž๐›ผโˆ’1(๐‘ก,๐‘ ).(2.35) The case ๐‘šโ‰ฅ2 can be verified by the induction principle.

We note that an extension of this property for derivatives of noninteger orders will be performed in Section 4.

Now we can continue with the introduction of (๐‘ž,โ„Ž)-fractional integral and derivative of a function ๎‚๐•‹๐‘“โˆถ๐‘Ž(๐‘ž,โ„Ž)โ†’โ„. Let ๎‚๐•‹๐‘กโˆˆ๐‘Ž(๐‘ž,โ„Ž). Our previous considerations (in particular, the Cauchy formula (2.5) along with the relations (2.22) and (2.29)) warrant us to introduce the nabla (๐‘ž,โ„Ž)-fractional integral of order ๐›ผโˆˆโ„+ over the time scale interval ๎‚๐•‹[๐‘Ž,๐‘ก]โˆฉ๐‘Ž(๐‘ž,โ„Ž) as ๐‘Žโˆ‡โˆ’๐›ผ(๐‘ž,โ„Ž)๎€œ๐‘“(๐‘ก)=๐‘ก๐‘Ž๎โ„Ž๐›ผโˆ’1(๐‘ก,๐œŒ(๐œ))๐‘“(๐œ)โˆ‡๐œ(2.36) (see also [7]). The nabla (๐‘ž,โ„Ž)-fractional derivative of order ๐›ผโˆˆโ„+ is then defined by ๐‘Žโˆ‡๐›ผ(๐‘ž,โ„Ž)๐‘“(๐‘ก)=โˆ‡๐‘š๐‘Ž(๐‘ž,โ„Ž)โˆ‡โˆ’(๐‘šโˆ’๐›ผ)(๐‘ž,โ„Ž)๐‘“(๐‘ก),(2.37) where ๐‘šโˆˆโ„ค+ is given by ๐‘šโˆ’1<๐›ผโ‰ค๐‘š. For the sake of completeness, we put ๐‘Žโˆ‡0(๐‘ž,โ„Ž)๐‘“(๐‘ก)=๐‘“(๐‘ก).(2.38)

As we noted earlier, a reasonable introduction of fractional integrals and fractional derivatives on arbitrary time scales remains an open problem. In the previous part, we have consistently used (and in the sequel, we shall consistently use) the time scale notation of main procedures and operations to outline a possible way out to further generalizations.

3. A Linear Initial Value Problem

In this section, we are going to discuss the linear initial value problem ๐‘š๎“๐‘—=1๐‘๐‘šโˆ’๐‘—+1(๐‘ก)๐‘Žโˆ‡๐›ผโˆ’๐‘—+1(๐‘ž,โ„Ž)๐‘ฆ(๐‘ก)+๐‘0๎‚๐•‹(๐‘ก)๐‘ฆ(๐‘ก)=0,๐‘กโˆˆ๐œŽ๐‘š+1(๐‘Ž)(๐‘ž,โ„Ž),(3.1)๐‘Žโˆ‡๐›ผโˆ’๐‘—(๐‘ž,โ„Ž)|||๐‘ฆ(๐‘ก)๐‘ก=๐œŽ๐‘š(๐‘Ž)=๐‘ฆ๐›ผโˆ’๐‘—,๐‘—=1,2,โ€ฆ,๐‘š,(3.2) where ๐›ผโˆˆโ„+ and ๐‘šโˆˆโ„ค+ are such that ๐‘šโˆ’1<๐›ผโ‰ค๐‘š. Further, we assume that ๐‘๐‘—(๐‘ก) are arbitrary real-valued functions on ๎‚๐•‹๐œŽ๐‘š+1(๐‘Ž)(๐‘ž,โ„Ž)(๐‘—=1,โ€ฆ,๐‘šโˆ’1), ๐‘๐‘š(๐‘ก)=1 on ๎‚๐•‹๐œŽ๐‘š+1(๐‘Ž)(๐‘ž,โ„Ž) and ๐‘ฆ๐›ผโˆ’๐‘—(๐‘—=1,โ€ฆ,๐‘š) are arbitrary real scalars.

If ๐›ผ is a positive integer, then (3.1)-(3.2) becomes the standard discrete initial value problem. If ๐›ผ is not an integer, then applying the definition of nabla (๐‘ž,โ„Ž)-fractional derivatives, we can observe that (3.1) is of the general form ๐‘›โˆ’1๎“๐‘–=0๐‘Ž๐‘–๎€ท๐œŒ(๐‘ก)๐‘ฆ๐‘–๎€ธ๎‚๐•‹(๐‘ก)=0,๐‘กโˆˆ๐œŽ๐‘š+1(๐‘Ž)(๐‘ž,โ„Ž),๐‘›beingsuchthat๐‘ก=๐œŽ๐‘›(๐‘Ž),(3.3) which is usually referred to as the equation of Volterra type. If such an equation has two different solutions, then their values differ at least at one of the points ๐œŽ(๐‘Ž),๐œŽ2(๐‘Ž),โ€ฆ,๐œŽ๐‘š(๐‘Ž). In particular, if ๐‘Ž0(๐‘ก)โ‰ 0 for all ๎‚๐•‹๐‘กโˆˆ๐œŽ๐‘š+1(๐‘Ž)(๐‘ž,โ„Ž), then arbitrary values of ๐‘ฆ(๐œŽ(๐‘Ž)),๐‘ฆ(๐œŽ2(๐‘Ž)),โ€ฆ,๐‘ฆ(๐œŽ๐‘š(๐‘Ž)) determine uniquely the solution ๐‘ฆ(๐‘ก) for all ๎‚๐•‹๐‘กโˆˆ๐œŽ๐‘š+1(๐‘Ž)(๐‘ž,โ„Ž). We show that the values ๐‘ฆ๐›ผโˆ’1,๐‘ฆ๐›ผโˆ’2,โ€ฆ,๐‘ฆ๐›ผโˆ’๐‘š, introduced by (3.2), keep the same properties.

Proposition 3.1. Let ๎‚๐•‹๐‘ฆโˆถ๐œŽ(๐‘Ž)(๐‘ž,โ„Ž)โ†’โ„ be a function. Then (3.2) represents a one-to-one mapping between the vectors (๐‘ฆ(๐œŽ(๐‘Ž)),๐‘ฆ(๐œŽ2(๐‘Ž)),โ€ฆ,๐‘ฆ(๐œŽ๐‘š(๐‘Ž))) and (๐‘ฆ๐›ผโˆ’1,๐‘ฆ๐›ผโˆ’2,โ€ฆ,๐‘ฆ๐›ผโˆ’๐‘š).

Proof. The case ๐›ผโˆˆโ„ค+ is well known from the literature. Let ๐›ผโˆ‰โ„ค+. We wish to show that the values of ๐‘ฆ(๐œŽ(๐‘Ž)),๐‘ฆ(๐œŽ2(๐‘Ž)),โ€ฆ,๐‘ฆ(๐œŽ๐‘š(๐‘Ž)) determine uniquely the values of ๐‘Žโˆ‡๐›ผโˆ’1(๐‘ž,โ„Ž)||๐‘ฆ(๐‘ก)๐‘ก=๐œŽ๐‘š(๐‘Ž),๐‘Žโˆ‡๐›ผโˆ’2(๐‘ž,โ„Ž)||๐‘ฆ(๐‘ก)๐‘ก=๐œŽ๐‘š(๐‘Ž),โ€ฆ,๐‘Žโˆ‡๐›ผโˆ’๐‘š(๐‘ž,โ„Ž)||๐‘ฆ(๐‘ก)๐‘ก=๐œŽ๐‘š(๐‘Ž)(3.4) and vice versa. Utilizing the relation ๐‘Žโˆ‡๐›ผโˆ’๐‘—(๐‘ž,โ„Ž)|||๐‘ฆ(๐‘ก)๐‘ก=๐œŽ๐‘š(๐‘Ž)=๐‘š๎“๐‘˜=1๐œˆ๎€ท๐œŽ๐‘šโˆ’๐‘˜+1๎€ธ๎โ„Ž(๐‘Ž)๐‘—โˆ’1โˆ’๐›ผ๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽ๐‘šโˆ’๐‘˜๎€ธ๐‘ฆ๎€ท๐œŽ(๐‘Ž)๐‘šโˆ’๐‘˜+1๎€ธ(๐‘Ž)(3.5) (see [7, Propositionsโ€‰โ€‰1 andโ€‰โ€‰3] with respect to (2.30)), we can rewrite (3.2) as the linear mapping ๐‘š๎“๐‘˜=1๐‘Ÿ๐‘—๐‘˜๐‘ฆ๎€ท๐œŽ๐‘šโˆ’๐‘˜+1๎€ธ(๐‘Ž)=๐‘ฆ๐›ผโˆ’๐‘—,๐‘—=1,โ€ฆ,๐‘š,(3.6) where ๐‘Ÿ๐‘—๐‘˜๎€ท๐œŽ=๐œˆ๐‘šโˆ’๐‘˜+1๎€ธ๎โ„Ž(๐‘Ž)๐‘—โˆ’1โˆ’๐›ผ๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽ๐‘šโˆ’๐‘˜๎€ธ(๐‘Ž),๐‘—,๐‘˜=1,โ€ฆ,๐‘š(3.7) are elements of the transformation matrix ๐‘…๐‘š. We show that ๐‘…๐‘š is regular. Obviously, det๐‘…๐‘š=๎ƒฉ๐‘š๎‘๐‘˜=1๐œˆ๎€ท๐œŽ๐‘˜๎€ธ๎ƒช(๐‘Ž)det๐ป๐‘š,(3.8) where ๐ป๐‘š=โŽ›โŽœโŽœโŽœโŽœโŽ๎โ„Žโˆ’๐›ผ๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽ๐‘šโˆ’1๎€ธ๎โ„Ž(๐‘Ž)โˆ’๐›ผ๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽ๐‘šโˆ’2๎€ธโ‹ฏ๎โ„Ž(๐‘Ž)โˆ’๐›ผ(๐œŽ๐‘š๎โ„Ž(๐‘Ž),๐‘Ž)1โˆ’๐›ผ๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽ๐‘šโˆ’1๎€ธ๎โ„Ž(๐‘Ž)1โˆ’๐›ผ๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽ๐‘šโˆ’2๎€ธโ‹ฏ๎โ„Ž(๐‘Ž)1โˆ’๐›ผ(๐œŽ๐‘š๎โ„Ž(๐‘Ž),๐‘Ž)โ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘šโˆ’1โˆ’๐›ผ๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽ๐‘šโˆ’1๎€ธ๎โ„Ž(๐‘Ž)๐‘šโˆ’1โˆ’๐›ผ๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽ๐‘šโˆ’2๎€ธโ‹ฏ๎โ„Ž(๐‘Ž)๐‘šโˆ’1โˆ’๐›ผ(๐œŽ๐‘šโŽžโŽŸโŽŸโŽŸโŽŸโŽ (๐‘Ž),๐‘Ž).(3.9) To calculate det๐ป๐‘š, we employ some elementary operations preserving the value of det๐ป๐‘š. Using the properties ๎โ„Ž๐‘–โˆ’๐›ผ๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽโ„“๎€ธ(๐‘Ž)โˆ’๐œˆ(๐œŽ๐‘š๎โ„Ž(๐‘Ž))๐‘–โˆ’๐›ผโˆ’1๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽโ„“๎€ธ=๎โ„Ž(๐‘Ž)๐‘–โˆ’๐›ผ๎€ท๐œŽ๐‘šโˆ’1(๐‘Ž),๐œŽโ„“๎€ธ๎โ„Ž(๐‘Ž)(๐‘–=1,2,โ€ฆ,๐‘šโˆ’1,๐‘™=0,1,โ€ฆ๐‘šโˆ’2),๐‘–โˆ’๐›ผ๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽ๐‘šโˆ’1๎€ธ(๐‘Ž)โˆ’๐œˆ(๐œŽ๐‘š๎โ„Ž(๐‘Ž))๐‘–โˆ’๐›ผโˆ’1๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽ๐‘šโˆ’1๎€ธ(๐‘Ž)=0,(3.10) which follow from Lemma 2.3, we multiply the ๐‘–th row (๐‘–=1,2,โ€ฆ,๐‘šโˆ’1) of ๐ป๐‘š by โˆ’๐œˆ(๐œŽ๐‘š(๐‘Ž)) and add it to the successive one. We arrive at the form โŽ›โŽœโŽœโŽœโŽœโŽ๎โ„Žโˆ’๐›ผ๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽ๐‘šโˆ’1๎€ธ0โ‹ฎ0(๐‘Ž)๎โ„Žโˆ’๐›ผ๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽ๐‘šโˆ’2๎€ธโ‹ฏ๎โ„Ž(๐‘Ž)โˆ’๐›ผ(๐œŽ๐‘š(๐‘Ž),๐‘Ž)๐ป๐‘šโˆ’1โŽžโŽŸโŽŸโŽŸโŽŸโŽ .(3.11) Then we apply repeatedly this procedure to obtain the triangular matrix โŽ›โŽœโŽœโŽœโŽœโŽ๎โ„Žโˆ’๐›ผ๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽ๐‘šโˆ’1๎€ธ๎โ„Ž(๐‘Ž)โˆ’๐›ผ๎€ท๐œŽ๐‘š(๐‘Ž),๐œŽ๐‘šโˆ’2๎€ธโ‹ฏ๎โ„Ž(๐‘Ž)โˆ’๐›ผ(๐œŽ๐‘š0๎โ„Ž(๐‘Ž),๐‘Ž)1โˆ’๐›ผ๎€ท๐œŽ๐‘šโˆ’1(๐‘Ž),๐œŽ๐‘šโˆ’2๎€ธโ‹ฏ๎โ„Ž(๐‘Ž)1โˆ’๐›ผ๎€ท๐œŽ๐‘šโˆ’1๎€ธ๎โ„Ž(๐‘Ž),๐‘Žโ‹ฎโ‹ฎโ‹ฑโ‹ฎ00โ‹ฏ๐‘šโˆ’1โˆ’๐›ผโŽžโŽŸโŽŸโŽŸโŽŸโŽ (๐œŽ(๐‘Ž),๐‘Ž).(3.12) Since ๎โ„Ž๐‘–โˆ’๐›ผ(๐œŽ๐‘˜(๐‘Ž),๐œŽ๐‘˜โˆ’1(๐‘Ž))=(๐œˆ(๐œŽ๐‘˜(๐‘Ž))๐‘–โˆ’๐›ผ(๐‘–=0,1,โ€ฆ,๐‘šโˆ’1), we get det๐ป๐‘š=๐‘š๎‘๐‘˜=1๎€ท๐œˆ๎€ท๐œŽ๐‘˜(๐‘Ž)๎€ธ๎€ธ๐‘šโˆ’๐‘˜โˆ’๐›ผ,thatis,det๐‘…๐‘š=๐‘š๎‘๐‘˜=1๎€ท๐œˆ๎€ท๐œŽ๐‘˜(๐‘Ž)๎€ธ๎€ธ๐‘šโˆ’๐‘˜โˆ’๐›ผ+1โ‰ 0.(3.13) Thus the matrix ๐‘…๐‘š is regular, hence the corresponding mapping (3.6) is one to one.

Now we approach a problem of the existence and uniqueness of (3.1)-(3.2). First we recall the general notion of ๐œˆ-regressivity of a matrix function and a corresponding linear nabla dynamic system (see [9]).

Definition 3.2. An ๐‘›ร—๐‘›-matrix-valued function ๐ด(๐‘ก) on a time scale ๐•‹ is called ๐œˆ-regressive provided det(๐ผโˆ’๐œˆ(๐‘ก)๐ด(๐‘ก))โ‰ 0โˆ€๐‘กโˆˆ๐•‹๐œ…,(3.14) where ๐ผ is the identity matrix. Further, we say that the linear dynamic system โˆ‡๐‘ง(๐‘ก)=๐ด(๐‘ก)๐‘ง(๐‘ก)(3.15) is ๐œˆ-regressive provided that ๐ด(๐‘ก) is ๐œˆ-regressive.

Considering a higher order linear difference equation, the notion of ๐œˆ-regressivity for such an equation can be introduced by means of its transformation to the corresponding first order linear dynamic system. We are going to follow this approach and generalize the notion of ๐œˆ-regressivity for the linear fractional difference equation (3.1).

Definition 3.3. Let ๐›ผโˆˆโ„+ and ๐‘šโˆˆโ„ค+ be such that ๐‘šโˆ’1<๐›ผโ‰ค๐‘š. Then (3.1) is called ๐œˆ-regressive provided the matrix โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽโˆ’๐‘๐ด(๐‘ก)=010โ‹ฏ0001โ‹ฑโ‹ฎโ‹ฎโ‹ฎโ‹ฑโ‹ฑ000โ‹ฏ010(๐‘ก)๐œˆ๐‘šโˆ’๐›ผ(๐‘ก)โˆ’๐‘1(๐‘ก)โ‹ฏโˆ’๐‘๐‘šโˆ’2(๐‘ก)โˆ’๐‘๐‘šโˆ’1โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ (๐‘ก)(3.16) is ๐œˆ-regressive.

Remark 3.4. The explicit expression of the ๐œˆ-regressivity property for (3.1) can be read as 1+๐‘šโˆ’1๎“๐‘—=1๐‘๐‘šโˆ’๐‘—(๐‘ก)(๐œˆ(๐‘ก))๐‘—+๐‘0(๐‘ก)(๐œˆ(๐‘ก))๐›ผ๎‚๐•‹โ‰ 0โˆ€๐‘กโˆˆ๐œŽ๐‘š+1(๐‘Ž)(๐‘ž,โ„Ž).(3.17) If ๐›ผ is a positive integer, then both these introductions agree with the definition of ๐œˆ-regressivity of a higher order linear difference equation presented in [9].

Theorem 3.5. Let (3.1) be ๐œˆ-regressive. Then the problem (3.1)-(3.2) has a unique solution defined for all ๎‚๐•‹๐‘กโˆˆ๐œŽ(๐‘Ž)(๐‘ž,โ„Ž).

Proof. The conditions (3.2) enable us to determine the values of ๐‘ฆ(๐œŽ(๐‘Ž)),๐‘ฆ(๐œŽ2(๐‘Ž)),โ€ฆ,๐‘ฆ(๐œŽ๐‘š(๐‘Ž)) by the use of (3.6). To calculate the values of ๐‘ฆ(๐œŽ๐‘š+1(๐‘Ž)),๐‘ฆ(๐œŽ๐‘š+2(๐‘Ž)),โ€ฆ, we perform the transformation ๐‘ง๐‘—(๐‘ก)=๐‘Žโˆ‡๐›ผโˆ’๐‘š+๐‘—โˆ’1(๐‘ž,โ„Ž)๎‚๐•‹๐‘ฆ(๐‘ก),๐‘กโˆˆ๐œŽ๐‘—(๐‘Ž)(๐‘ž,โ„Ž),๐‘—=1,2,โ€ฆ,๐‘š(3.18) which allows us to rewrite (3.1) into a matrix form. Before doing this, we need to express ๐‘ฆ(๐‘ก) in terms of ๐‘ง1(๐‘ก),๐‘ง1(๐œŒ(๐‘ก)),โ€ฆ,๐‘ง1(๐œŽ(๐‘Ž)). Applying the relation ๐‘Žโˆ‡๐‘Ž๐‘šโˆ’๐›ผ(๐‘ž,โ„Ž)โˆ‡โˆ’(๐‘šโˆ’๐›ผ)(๐‘ž,โ„Ž)๐‘ฆ(๐‘ก)=๐‘ฆ(๐‘ก) (see [7]) and expanding the fractional derivative, we arrive at ๐‘ฆ(๐‘ก)=๐‘Žโˆ‡๐‘šโˆ’๐›ผ(๐‘ž,โ„Ž)๐‘ง1๐‘ง(๐‘ก)=1(๐‘ก)๐œˆ๐‘šโˆ’๐›ผ+๎€œ(๐‘ก)๐‘Ž๐œŒ(๐‘ก)๎โ„Ž๐›ผโˆ’๐‘šโˆ’1(๐‘ก,๐œŒ(๐œ))๐‘ง1(๐œ)โˆ‡๐œ.(3.19) Therefore, the problem (3.1)-(3.2) can be rewritten to the vector form ๐‘Žโˆ‡(๐‘ž,โ„Ž)๎‚๐•‹๐‘ง(๐‘ก)=๐ด(๐‘ก)๐‘ง(๐‘ก)+๐‘(๐‘ก),๐‘กโˆˆ๐œŽ๐‘š+1(๐‘Ž)(๐‘ž,โ„Ž),๐‘ง(๐œŽ๐‘š๎€ท๐‘ฆ(๐‘Ž))=๐›ผโˆ’๐‘š,โ€ฆ,๐‘ฆ๐›ผโˆ’1๎€ธ๐‘‡,(3.20) where ๎€ท๐‘ง๐‘ง(๐‘ก)=1(๐‘ก),โ€ฆ,๐‘ง๐‘š๎€ธ(๐‘ก)๐‘‡๎‚ต,๐‘(๐‘ก)=0,โ€ฆ,0,โˆ’๐‘0๎€œ(๐‘ก)๐‘Ž๐œŒ(๐‘ก)๎โ„Ž๐›ผโˆ’๐‘šโˆ’1(๐‘ก,๐œŒ(๐œ))๐‘ง1๎‚ถ(๐œ)โˆ‡๐œ๐‘‡(3.21) and ๐ด(๐‘ก) is given by (3.16). The ๐œˆ-regressivity of the matrix ๐ด(๐‘ก) enables us to write ๐‘ง(๐‘ก)=(๐ผโˆ’๐œˆ(๐‘ก)๐ด(๐‘ก))โˆ’1๎‚๐•‹(๐‘ง(๐œŒ(๐‘ก))+๐œˆ(๐‘ก)๐‘(๐‘ก)),๐‘กโˆˆ๐œŽ๐‘š+1(๐‘Ž)(๐‘ž,โ„Ž),(3.22) hence, using the value of ๐‘ง(๐œŽ๐‘š(๐‘Ž)), we can solve this system by the step method starting from ๐‘ก=๐œŽ๐‘š+1(๐‘Ž). The solution ๐‘ฆ(๐‘ก) of the original initial value problem (3.1)-(3.2) is then given by the formula (3.19).

Remark 3.6. The previous assertion on the existence and uniqueness of the solution can be easily extended to the initial value problem involving nonhomogeneous linear equations as well as some nonlinear equations.

The final goal of this section is to investigate the structure of the solutions of (3.1). We start with the following notion.

Definition 3.7. Let ๐›พโˆˆโ„, 0โ‰ค๐›พ<1. For ๐‘š functions ๐‘ฆ๐‘—โˆถ๎‚๐•‹๐‘Ž(๐‘ž,โ„Ž)โ†’โ„(๐‘—=1,2,โ€ฆ,๐‘š), we define the ๐›พ-Wronskian ๐‘Š๐›พ(๐‘ฆ1,โ€ฆ,๐‘ฆ๐‘š)(๐‘ก) as determinant of the matrix ๐‘‰๐›พ๎€ท๐‘ฆ1,โ€ฆ,๐‘ฆ๐‘š๎€ธโŽ›โŽœโŽœโŽœโŽœโŽ(๐‘ก)=๐‘Žโˆ‡โˆ’๐›พ(๐‘ž,โ„Ž)๐‘ฆ1(๐‘ก)๐‘Žโˆ‡โˆ’๐›พ(๐‘ž,โ„Ž)๐‘ฆ2(๐‘ก)โ‹ฏ๐‘Žโˆ‡โˆ’๐›พ(๐‘ž,โ„Ž)๐‘ฆ๐‘š(๐‘ก)๐‘Žโˆ‡1โˆ’๐›พ(๐‘ž,โ„Ž)๐‘ฆ1(๐‘ก)๐‘Žโˆ‡1โˆ’๐›พ(๐‘ž,โ„Ž)๐‘ฆ2(๐‘ก)โ‹ฏ๐‘Žโˆ‡1โˆ’๐›พ(๐‘ž,โ„Ž)๐‘ฆ๐‘š(๐‘ก)โ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘Žโˆ‡๐‘šโˆ’1โˆ’๐›พ(๐‘ž,โ„Ž)๐‘ฆ1(๐‘ก)๐‘Žโˆ‡๐‘šโˆ’1โˆ’๐›พ(๐‘ž,โ„Ž)๐‘ฆ2(๐‘ก)โ‹ฏ๐‘Žโˆ‡๐‘šโˆ’1โˆ’๐›พ(๐‘ž,โ„Ž)๐‘ฆ๐‘šโŽžโŽŸโŽŸโŽŸโŽŸโŽ ๎‚๐•‹(๐‘ก),๐‘กโˆˆ๐œŽ๐‘š(๐‘Ž)(๐‘ž,โ„Ž).(3.23)

Remark 3.8. Note that the first row of this matrix involves fractional order integrals. It is a consequence of the form of initial conditions utilized in our investigations. Of course, this introduction of ๐‘Š๐›พ(๐‘ฆ1,โ€ฆ,๐‘ฆ๐‘š)(๐‘ก) coincides for ๐›พ=0 with the classical definition of the Wronskian (see [8]). Moreover, it holds ๐‘Š๐›พ(๐‘ฆ1,โ€ฆ,๐‘ฆ๐‘š)(๐‘ก)=๐‘Š0(๐‘Žโˆ‡โˆ’๐›พ(๐‘ž,โ„Ž)๐‘ฆ1,โ€ฆ,๐‘Žโˆ‡โˆ’๐›พ(๐‘ž,โ„Ž)๐‘ฆ๐‘š)(๐‘ก).

Theorem 3.9. Let functions ๐‘ฆ1(๐‘ก),โ€ฆ,๐‘ฆ๐‘š(๐‘ก) be solutions of the ๐œˆ-regressive equation (3.1) and let ๐‘Š๐‘šโˆ’๐›ผ(๐‘ฆ1,โ€ฆ,๐‘ฆ๐‘š)(๐œŽ๐‘š(๐‘Ž))โ‰ 0. Then any solution ๐‘ฆ(๐‘ก) of (3.1) can be written in the form ๐‘ฆ(๐‘ก)=๐‘š๎“๐‘˜=1๐‘๐‘˜๐‘ฆ๐‘˜๎‚๐•‹(๐‘ก),๐‘กโˆˆ๐œŽ(๐‘Ž)(๐‘ž,โ„Ž),(3.24) where ๐‘1,โ€ฆ,๐‘๐‘š are real constants.

Proof. Let ๐‘ฆ(๐‘ก) be a solution of (3.1). By Proposition 3.1, there exist real scalars ๐‘ฆ๐›ผโˆ’1,โ€ฆ,๐‘ฆ๐›ผโˆ’๐‘š such that ๐‘ฆ(๐‘ก) is satisfying (3.2). Now we consider the function โˆ‘๐‘ข(๐‘ก)=๐‘š๐‘˜=1๐‘๐‘˜๐‘ฆ๐‘˜(๐‘ก), where the ๐‘š-tuple (๐‘1,โ€ฆ,๐‘๐‘š) is the unique solution of ๐‘‰๐‘šโˆ’๐›ผ๎€ท๐‘ฆ1,โ€ฆ,๐‘ฆ๐‘š๎€ธ(๐œŽ๐‘šโŽ›โŽœโŽœโŽœโŽ๐‘(๐‘Ž))โ‹…1๐‘2โ‹ฎ๐‘๐‘šโŽžโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽ๐‘ฆ๐›ผโˆ’๐‘š๐‘ฆ๐›ผโˆ’๐‘š+1โ‹ฎ๐‘ฆ๐›ผโˆ’1โŽžโŽŸโŽŸโŽŸโŽ .(3.25) The linearity of (3.1) implies that ๐‘ข(๐‘ก) has to be its solution. Moreover, it holds ๐‘Žโˆ‡๐›ผโˆ’๐‘—(๐‘ž,โ„Ž)|||๐‘ข(๐‘ก)๐‘ก=๐œŽ๐‘š(๐‘Ž)=๐‘ฆ๐›ผโˆ’๐‘—,๐‘—=1,2,โ€ฆ,๐‘š,(3.26) hence ๐‘ข(๐‘ก) is a solution of the initial value problem (3.1)-(3.2). By Theorem 3.5, it must be ๐‘ฆ(๐‘ก)=๐‘ข(๐‘ก) for all ๎‚๐•‹๐‘กโˆˆ๐œŽ(๐‘Ž)(๐‘ž,โ„Ž) and (3.24) holds.

Remark 3.10. The formula (3.24) is essentially an expression of the general solution of (3.1).

4. Two-Term Equation and (๐‘ž,โ„Ž)-Mittag-Leffler Function

Our main interest in this section is to find eigenfunctions of the fractional operator ๐‘Žโˆ‡๐›ผ(๐‘ž,โ„Ž), ๐›ผโˆˆโ„+. In other words, we wish to solve (3.1) in a special form ๐‘Žโˆ‡๐›ผ(๐‘ž,โ„Ž)๎‚๐•‹๐‘ฆ(๐‘ก)=๐œ†๐‘ฆ(๐‘ก),๐œ†โˆˆโ„,๐‘กโˆˆ๐œŽ๐‘š+1(๐‘Ž)(๐‘ž,โ„Ž).(4.1) Throughout this section, we assume that ๐œˆ-regressivity condition for (4.1) is ensured, that is, ๐œ†(๐œˆ(๐‘ก))๐›ผโ‰ 1.(4.2)

Discussions on methods of solving fractional difference equations are just at the beginning. Some techniques how to explicitly solve these equations (at least in particular cases) are exhibited, for example, in [12โ€“14], where a discrete analogue of the Laplace transform turns out to be the most developed method. In this section, we describe the technique not utilizing the transform method, but directly originating from the role which is played by the Mittag-Leffler function in the continuous fractional calculus (see, e.g., [15]). In particular, we introduce the notion of a discrete Mittag-Leffler function in a setting formed by the time scale ๎‚๐•‹๐‘Ž(๐‘ž,โ„Ž) and demonstrate its significance with respect to eigenfunctions of the operator ๐‘Žโˆ‡๐›ผ(๐‘ž,โ„Ž). These results generalize and extend those derived in [16, 17].

We start with the power rule stated in Lemma 2.3 and perform its extension to fractional integrals and derivatives.

Proposition 4.1. Let ๐›ผโˆˆโ„+, ๐›ฝโˆˆโ„ and ๎‚๐•‹๐‘กโˆˆ๐œŽ(๐‘Ž)(๐‘ž,โ„Ž). Then it holds ๐‘Žโˆ‡โˆ’๐›ผ(๐‘ž,โ„Ž)๎โ„Ž๐›ฝ๎โ„Ž(๐‘ก,๐‘Ž)=๐›ผ+๐›ฝ(๐‘ก,๐‘Ž).(4.3)

Proof. Let ๎‚๐•‹๐‘กโˆˆ๐œŽ(๐‘Ž)(๐‘ž,โ„Ž) be such that ๐‘ก=๐œŽ๐‘›(๐‘Ž) for some ๐‘›โˆˆโ„ค+. We have ๐‘Žโˆ‡โˆ’๐›ผ(๐‘ž,โ„Ž)๎โ„Ž๐›ฝ(๐‘ก,๐‘Ž)=๐‘›โˆ’1๎“๐‘˜=0๎โ„Ž๐›ผโˆ’1๎€ท๐‘ก,๐œŒ๐‘˜+1๎€ธ๐œˆ๎€ท๐œŒ(๐‘ก)๐‘˜๎€ธ๎โ„Ž(๐‘ก)๐›ฝ๎€ท๐œŒ๐‘˜๎€ธ=(๐‘ก),๐‘Ž๐‘›โˆ’1๎“๐‘˜=0(๐œˆ(๐‘ก))๐›ผโˆ’1๎‚ธ๐‘˜๎‚นโˆ’๐›ผฬƒ๐‘ž(โˆ’1)๐‘˜ฬƒ๐‘ž๎‚€2๎‚(๐›ผโˆ’1)๐‘˜+๐‘˜+1ฬƒ๐‘ž๐‘˜ร—๎€ท๐œˆ๎€ท๐œŒ๐œˆ(๐‘ก)๐‘˜(๐‘ก)๎€ธ๎€ธ๐›ฝ๎‚ธ๎‚นโˆ’๐›ฝโˆ’1๐‘›โˆ’๐‘˜โˆ’1ฬƒ๐‘ž(โˆ’1)๐‘›โˆ’๐‘˜โˆ’1ฬƒ๐‘ž๎‚€2๎‚๐›ฝ(๐‘›โˆ’๐‘˜โˆ’1)+๐‘›โˆ’๐‘˜=(๐œˆ(๐‘ก))๐›ผ+๐›ฝ๐‘›โˆ’1๎“๐‘˜=0๎‚ธ๐‘˜๎‚นโˆ’๐›ผฬƒ๐‘ž๎‚ธ๎‚นโˆ’๐›ฝโˆ’1๐‘›โˆ’๐‘˜โˆ’1ฬƒ๐‘ž(โˆ’1)๐‘›โˆ’1ฬƒ๐‘ž๐‘˜2๎€ท๐‘›2๎€ธโˆ’๐‘˜(๐‘›โˆ’1)+๐‘˜๐›ผ++๐›ฝ(๐‘›โˆ’1)=(๐œˆ(๐‘ก))๐›ผ+๐›ฝ๐‘›โˆ’1๎“๐‘˜=0๎‚ธ๎‚นโˆ’๐›ผ๐‘›โˆ’๐‘˜โˆ’1ฬƒ๐‘ž๎‚ธ๐‘˜๎‚นโˆ’๐›ฝโˆ’1ฬƒ๐‘žร—(โˆ’1)๐‘›โˆ’1ฬƒ๐‘ž(๐‘›โˆ’๐‘˜โˆ’1)2๎€ท๐‘›2๎€ธโˆ’(๐‘›โˆ’๐‘˜โˆ’1)(๐‘›โˆ’1)+(๐‘›โˆ’๐‘˜โˆ’1)๐›ผ++๐›ฝ(๐‘›โˆ’1)=(๐œˆ(๐‘ก))๐›ผ+๐›ฝ๐‘›โˆ’1๎“๐‘˜=0๎‚ธ๎‚นโˆ’๐›ผ๐‘›โˆ’๐‘˜โˆ’1ฬƒ๐‘ž๎‚ธ๐‘˜๎‚นโˆ’๐›ฝโˆ’1ฬƒ๐‘ž(โˆ’1)๐‘›โˆ’1ฬƒ๐‘ž๐‘˜2๎€ท๐‘›2๎€ธโˆ’๐‘˜(๐‘›โˆ’1)โˆ’๐‘˜๐›ผ+(๐›ผ+๐›ฝ)(๐‘›โˆ’1)+=(๐œˆ(๐‘ก))๐›ผ+๐›ฝ๎‚ธ๎‚นโˆ’๐›ผโˆ’๐›ฝโˆ’1๐‘›โˆ’1ฬƒ๐‘ž(โˆ’1)๐‘›โˆ’1ฬƒ๐‘ž๎€ท๐‘›2๎€ธ(๐›ผ+๐›ฝ)(๐‘›โˆ’1)+=๎โ„Ž๐›ผ+๐›ฝ(๐‘ก,๐‘Ž),(4.4) where we have used (2.30) on the second line and (2.17) on the last line.

Corollary 4.2. Let ๐›ผโˆˆโ„+, ๐›ฝโˆˆโ„, ๎‚๐•‹๐‘กโˆˆ๐œŽ๐‘š+1(๐‘Ž)(๐‘ž,โ„Ž), where ๐‘šโˆˆโ„ค+ is satisfying ๐‘šโˆ’1<๐›ผโ‰ค๐‘š. Then ๐‘Žโˆ‡๐›ผ(๐‘ž,โ„Ž)๎โ„Ž๐›ฝ๎ƒฏ๎โ„Ž(๐‘ก,๐‘Ž)=๐›ฝโˆ’๐›ผ(๐‘ก,๐‘Ž),๐›ฝโˆ’๐›ผโˆ‰{โˆ’1,โ€ฆ,โˆ’๐‘š},0,๐›ฝโˆ’๐›ผโˆˆ{โˆ’1,โ€ฆ,โˆ’๐‘š}.(4.5)

Proof. Proposition 4.1 implies that ๐‘Žโˆ‡๐›ผ(๐‘ž,โ„Ž)๎โ„Ž๐›ฝ(๐‘ก,๐‘Ž)=โˆ‡๐‘š(๐‘ž,โ„Ž)๎‚€๐‘Žโˆ‡โˆ’(๐‘šโˆ’๐›ผ)(๐‘ž,โ„Ž)๎โ„Ž๐›ฝ๎‚(๐‘ก,๐‘Ž)=โˆ‡๐‘š(๐‘ž,โ„Ž)๎โ„Ž๐‘š+๐›ฝโˆ’๐›ผ(๐‘ก,๐‘Ž).(4.6) Then the statement is an immediate consequence of Lemma 2.3.

Now we are in a position to introduce a (๐‘ž,โ„Ž)-discrete analogue of the Mittag-Leffler function. We recall that this function is essentially a generalized exponential function, and its two-parameter form (more convenient in the fractional calculus) can be introduced for ๐•‹=โ„ by the series expansion ๐ธ๐›ผ,๐›ฝ(๐‘ก)=โˆž๎“๐‘˜=0๐‘ก๐‘˜ฮ“(๐›ผ๐‘˜+๐›ฝ),๐›ผ,๐›ฝโˆˆโ„+,๐‘กโˆˆโ„.(4.7) The fractional calculus frequently employs (4.7), because the function ๐‘ก๐›ฝโˆ’1๐ธ๐›ผ,๐›ฝ(๐œ†๐‘ก๐›ผ)=โˆž๎“๐‘˜=0๐œ†๐‘˜๐‘ก๐›ผ๐‘˜+๐›ฝโˆ’1ฮ“(๐›ผ๐‘˜+๐›ฝ)(4.8) (a modified Mittag-Leffler function, see [15]) satisfies under special choices of ๐›ฝ a continuous (differential) analogy of (4.1). Some extensions of the definition formula (4.7) and their utilization in special fractional calculus operators can be found in [18, 19].

Considering the discrete calculus, the form (4.8) seems to be much more convenient for discrete extensions than the form (4.7), which requires, among others, the validity of the law of exponents. The following introduction extends the discrete Mittag-Leffler function defined and studied in [20] for the case ๐‘ž=โ„Ž=1.

Definition 4.3. Let ๐›ผ,๐›ฝ,๐œ†โˆˆโ„. We introduce the (๐‘ž,โ„Ž)-Mittag-Leffler function ๐ธ๐‘ ,๐œ†๐›ผ,๐›ฝ(๐‘ก) by the series expansion ๐ธ๐‘ ,๐œ†๐›ผ,๐›ฝ(๐‘ก)=โˆž๎“๐‘˜=0๐œ†๐‘˜๎โ„Ž๐›ผ๐‘˜+๐›ฝโˆ’1โŽ›โŽœโŽœโŽ=(๐‘ก,๐‘ )โˆž๎“๐‘˜=0๐œ†๐‘˜(๐‘กโˆ’๐‘ )((๐›ผ๐‘˜+๐›ฝโˆ’1)ฬƒ๐‘ž,โ„Ž)ฮ“ฬƒ๐‘ž(โŽžโŽŸโŽŸโŽ ๎‚๐•‹๐›ผ๐‘˜+๐›ฝ),๐‘ ,๐‘กโˆˆ๐‘Ž(๐‘ž,โ„Ž),๐‘กโ‰ฅ๐‘ .(4.9)

It is easy to check that the series on the right-hand side converges (absolutely) if |๐œ†|(๐œˆ(๐‘ก))๐›ผ<1. As it might be expected, the particular (๐‘ž,โ„Ž)-Mittag-Leffler function ๐ธ๐‘Ž,๐œ†1,1(๐‘ก)=๐‘›โˆ’1๎‘๐‘˜=01๎€ท๐œŒ1โˆ’๐œ†๐œˆ๐‘˜๎€ธ(๐‘ก),(4.10) where ๐‘›โˆˆโ„ค+ satisfies ๐‘ก=๐œŽ๐‘›(๐‘Ž), is a solution of the equation โˆ‡(๐‘ž,โ„Ž)๎‚๐•‹๐‘ฆ(๐‘ก)=๐œ†๐‘ฆ(๐‘ก),๐‘กโˆˆ๐œŽ(๐‘Ž)(๐‘ž,โ„Ž),(4.11) that is, it is a discrete (๐‘ž,โ„Ž)-analogue of the exponential function.

The main properties of the (๐‘ž,โ„Ž)-Mittag-Leffler function are described by the following assertion.

Theorem 4.4. (i) Let ๐œ‚โˆˆโ„+ and ๎‚๐•‹๐‘กโˆˆ๐œŽ(๐‘Ž)(๐‘ž,โ„Ž). Then ๐‘Žโˆ‡โˆ’๐œ‚(๐‘ž,โ„Ž)๐ธ๐‘Ž,๐œ†๐›ผ,๐›ฝ(๐‘ก)=๐ธ๐‘Ž,๐œ†๐›ผ,๐›ฝ+๐œ‚(๐‘ก).(4.12)
(ii) Let ๐œ‚โˆˆโ„+, ๐‘šโˆˆโ„ค+ be such that ๐‘šโˆ’1<๐œ‚โ‰ค๐‘š and let ๐›ผ๐‘˜+๐›ฝโˆ’1โˆ‰{0,โˆ’1,โ€ฆ,โˆ’๐‘š+1} for all ๐‘˜โˆˆโ„ค+. If ๎‚๐•‹๐‘กโˆˆ๐œŽ๐‘š+1(๐‘Ž)(๐‘ž,โ„Ž), then ๐‘Žโˆ‡๐œ‚(๐‘ž,โ„Ž)๐ธ๐‘Ž,๐œ†๐›ผ,๐›ฝ๎ƒฏ๐ธ(๐‘ก)=๐‘Ž,๐œ†๐›ผ,๐›ฝโˆ’๐œ‚(๐‘ก),๐›ฝโˆ’๐œ‚โˆ‰{0,โˆ’1,โ€ฆ,โˆ’๐‘š+1},๐œ†๐ธ๐‘Ž,๐œ†๐›ผ,๐›ฝโˆ’๐œ‚+๐›ผ(๐‘ก),๐›ฝโˆ’๐œ‚โˆˆ{0,โˆ’1,โ€ฆ,โˆ’๐‘š+1}.(4.13)

Proof. The part (i) follows immediately from Proposition 4.1. Considering the part (ii), we can write ๐‘Žโˆ‡๐œ‚(๐‘ž,โ„Ž)๐ธ๐‘Ž,๐œ†๐›ผ,๐›ฝ(๐‘ก)=๐‘Žโˆ‡๐œ‚โˆž(๐‘ž,โ„Ž)๎“๐‘˜=0๐œ†๐‘˜๎โ„Ž๐›ผ๐‘˜+๐›ฝโˆ’1(๐‘ก,๐‘Ž)=โˆž๎“๐‘˜=0๐œ†๐‘˜๐‘Žโˆ‡๐œ‚(๐‘ž,โ„Ž)๎โ„Ž๐›ผ๐‘˜+๐›ฝโˆ’1(๐‘ก,๐‘Ž)(4.14) due to the absolute convergence property.
If ๐‘˜โˆˆโ„ค+, then Corollary 4.2 implies the relation ๐‘Žโˆ‡๐œ‚(๐‘ž,โ„Ž)๎โ„Ž๐›ผ๐‘˜+๐›ฝโˆ’1๎โ„Ž(๐‘ก,๐‘Ž)=๐›ผ๐‘˜+๐›ฝโˆ’๐œ‚โˆ’1(๐‘ก,๐‘Ž)(4.15) due to the assumption ๐›ผ๐‘˜+๐›ฝโˆ’1โˆ‰{0,โˆ’1,โ€ฆ,โˆ’๐‘š+1}. If ๐‘˜=0, then two possibilities may occur. If ๐›ฝโˆ’๐œ‚โˆ‰{0,โˆ’1,โ€ฆ,โˆ’๐‘š+1}, we get (4.15) with ๐‘˜=0 which implies the validity of (4.13)1. If ๐›ฝโˆ’๐œ‚โˆˆ{0,โˆ’1,โ€ฆ,โˆ’๐‘š+1}, the nabla (๐‘ž,โ„Ž)-fractional derivative of this term is zero and by shifting the index ๐‘˜, we obtain (4.13)2.

Corollary 4.5. Let ๐›ผโˆˆโ„+ and ๐‘šโˆˆโ„ค+ be such that ๐‘šโˆ’1<๐›ผโ‰ค๐‘š. Then the functions ๐ธ๐‘Ž,๐œ†๐›ผ,๐›ฝ(๐‘ก),๐›ฝ=๐›ผโˆ’๐‘š+1,โ€ฆ,๐›ผโˆ’1,๐›ผ(4.16) define eigenfunctions of the operator ๐‘Žโˆ‡๐›ผ(๐‘ž,โ„Ž) on each set ๎‚๐•‹[๐œŽ(๐‘Ž),๐‘]โˆฉ๐œŽ(๐‘Ž)(๐‘ž,โ„Ž), where ๎‚๐•‹๐‘โˆˆ๐œŽ(๐‘Ž)(๐‘ž,โ„Ž) is satisfying |๐œ†|(๐œˆ(๐‘))๐›ผ<1.

Proof. The assertion follows from Theorem 4.4 by the use of ๐œ‚=๐›ผ.

Our final aim is to show that any solution of (4.1) can be written as a linear combination of (๐‘ž,โ„Ž)-Mittag-Leffler functions (4.16).

Lemma 4.6. Let ๐›ผโˆˆโ„+ and ๐‘šโˆˆโ„ค+ be such that ๐‘šโˆ’1<๐›ผโ‰ค๐‘š. Then ๐‘Š๐‘šโˆ’๐›ผ๎‚€๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผโˆ’๐‘š+1,๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผโˆ’๐‘š+2,โ€ฆ,๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผ๎‚(๐œŽ๐‘š(๐‘Ž))=๐‘š๎‘๐‘˜=11๎€ท๐œˆ๎€ท๐œŽ1โˆ’๐œ†๐‘˜(๐‘Ž)๎€ธ๎€ธ๐›ผโ‰ 0.(4.17)

Proof. The case ๐‘š=1 is trivial. For ๐‘šโ‰ฅ2, we can formally write ๐œ†๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผโˆ’โ„“(๐‘ก)=๐ธ๐‘Ž,๐œ†๐›ผ,โˆ’โ„“(๐‘ก) for all ๎‚๐•‹๐‘กโˆˆ๐œŽ๐‘š(๐‘Ž)(๐‘ž,โ„Ž)(โ„“=0,โ€ฆ,๐‘šโˆ’2). Consequently, applying Theorem 4.4, the Wronskian can be expressed as ๐‘Š๐‘šโˆ’๐›ผ๎‚€๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผโˆ’๐‘š+1,๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผโˆ’๐‘š+2,โ€ฆ,๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผ๎‚(๐œŽ๐‘š(๐‘Ž))=det๐‘€๐‘š(๐œŽ๐‘š(๐‘Ž)),(4.18) where ๐‘€๐‘š(๐œŽ๐‘šโŽ›โŽœโŽœโŽœโŽœโŽ๐ธ(๐‘Ž))=๐‘Ž,๐œ†๐›ผ,1(๐œŽ๐‘š(๐‘Ž))๐ธ๐‘Ž,๐œ†๐›ผ,2(๐œŽ๐‘š(๐‘Ž))โ€ฆ๐ธ๐‘Ž,๐œ†๐›ผ,๐‘š(๐œŽ๐‘š๐ธ(๐‘Ž))๐‘Ž,๐œ†๐›ผ,0(๐œŽ๐‘š(๐‘Ž))๐ธ๐‘Ž,๐œ†๐›ผ,1(๐œŽ๐‘š(๐‘Ž))โ€ฆ๐ธ๐‘Ž,๐œ†๐›ผ,๐‘šโˆ’1(๐œŽ๐‘š๐ธ(๐‘Ž))โ€ฆโ€ฆโ‹ฑโ€ฆ๐‘Ž,๐œ†๐›ผ,2โˆ’๐‘š(๐œŽ๐‘š(๐‘Ž))๐ธ๐‘Ž,๐œ†๐›ผ,3โˆ’๐‘š(๐œŽ๐‘š(๐‘Ž))โ€ฆ๐ธ๐‘Ž,๐œ†๐›ผ,1(๐œŽ๐‘šโŽžโŽŸโŽŸโŽŸโŽŸโŽ (๐‘Ž)).(4.19) Using the ๐‘ž-Pascal rule (2.15), we obtain the equality ๐ธ๐‘Ž,๐œ†๐›ผ,๐‘–(๐œŽ๐‘š(๐‘Ž))โˆ’๐œˆ(๐œŽ(๐‘Ž))๐ธ๐‘Ž,๐œ†๐›ผ,๐‘–โˆ’1(๐œŽ๐‘š(๐‘Ž))=๐ธ๐œŽ(๐‘Ž),๐œ†๐›ผ,๐‘–(๐œŽ๐‘š(๐‘Ž)),๐‘–โˆˆโ„ค,๐‘–โ‰ฅ3โˆ’๐‘š.(4.20) Starting with the first row, (๐‘š2) elementary row operations of the type (4.20) transform the matrix ๐‘€๐‘š(๐œŽ๐‘š(๐‘Ž)) into the matrix ๎‚Š๐‘€๐‘š(๐œŽ๐‘šโŽ›โŽœโŽœโŽœโŽœโŽœโŽ๐ธ(๐‘Ž))=๐œŽ๐‘šโˆ’1(๐‘Ž),๐œ†๐›ผ,1(๐œŽ๐‘š(๐‘Ž))๐ธ๐œŽ๐‘šโˆ’1(๐‘Ž),๐œ†๐›ผ,2(๐œŽ๐‘š(๐‘Ž))โ€ฆ๐ธ๐œŽ๐‘šโˆ’1(๐‘Ž),๐œ†๐›ผ,๐‘š(๐œŽ๐‘š๐ธ(๐‘Ž))๐œŽ๐‘šโˆ’2(๐‘Ž),๐œ†๐›ผ,0(๐œŽ๐‘š(๐‘Ž))๐ธ๐œŽ๐‘šโˆ’2(๐‘Ž),๐œ†๐›ผ,1(๐œŽ๐‘š(๐‘Ž))โ€ฆ๐ธ๐œŽ๐‘šโˆ’2(๐‘Ž),๐œ†๐›ผ,๐‘šโˆ’1(๐œŽ๐‘š๐ธ(๐‘Ž))โ€ฆโ€ฆโ‹ฑโ€ฆ๐‘Ž,๐œ†๐›ผ,2โˆ’๐‘š(๐œŽ๐‘š(๐‘Ž))๐ธ๐‘Ž,๐œ†๐›ผ,3โˆ’๐‘š(๐œŽ๐‘š(๐‘Ž))โ€ฆ๐ธ๐‘Ž,๐œ†๐›ผ,1(๐œŽ๐‘šโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽ (๐‘Ž))(4.21) with the property ๎‚Š๐‘€det๐‘š(๐œŽ๐‘š(๐‘Ž))=det๐‘€๐‘š(๐œŽ๐‘š(๐‘Ž)). By Lemma 2.3, we have ๐ธ๐œŽ๐‘–(๐‘Ž),๐œ†๐›ผ,๐‘(๐œŽ๐‘š(๐‘Ž))โˆ’๐œˆ(๐œŽ๐‘š(๐‘Ž))๐ธ๐œŽ๐‘–(๐‘Ž),๐œ†๐›ผ,๐‘โˆ’1(๐œŽ๐‘š(๐‘Ž))=๐ธ๐œŽ๐‘–(๐‘Ž),๐œ†๐›ผ,๐‘๎€ท๐œŽ๐‘šโˆ’1๎€ธ๐ธ(๐‘Ž),๐‘–=0,โ€ฆ,๐‘šโˆ’2,๐œŽ๐‘–(๐‘Ž),๐œ†๐›ผ,๐‘(๐œŽ๐‘š(๐‘Ž))โˆ’๐œˆ(๐œŽ๐‘š(๐‘Ž))๐ธ๐œŽ๐‘–(๐‘Ž),๐œ†๐›ผ,๐‘โˆ’1(๐œŽ๐‘š(๐‘Ž))=0,๐‘–=๐‘šโˆ’1,(4.22) where ๐‘โˆˆโ„ค,๐‘โ‰ฅ3โˆ’๐‘š+๐‘–. Starting with the last column, using ๐‘šโˆ’1 elementary column operations of the type (4.22), we obtain the matrix โŽ›โŽœโŽœโŽœโŽœโŽœโŽ๐ธ๐œŽ๐‘šโˆ’1(๐‘Ž),๐œ†๐›ผ,1(๐œŽ๐‘š๐ธ(๐‘Ž))๐œŽ๐‘šโˆ’2(๐‘Ž),๐œ†๐›ผ,0(๐œŽ๐‘šโ‹ฎ๐ธ(๐‘Ž))๐‘Ž,๐œ†๐›ผ,2โˆ’๐‘š(๐œŽ๐‘š(๐‘Ž))0โ‹ฏ0๎‚Š๐‘€๐‘šโˆ’1๎€ท๐œŽ๐‘šโˆ’1(๎€ธ๐‘Ž)โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽ (4.23) preserving the value of ๎‚Š๐‘€det๐‘š(๐œŽ๐‘š(๐‘Ž)). Since ๐ธ๐œŽ๐‘šโˆ’1(๐‘Ž),๐œ†๐›ผ,1(๐œŽ๐‘š(๐‘Ž))=โˆž๎“๐‘˜=0๐œ†๐‘˜(๐œˆ(๐œŽ๐‘š(๐‘Ž)))๐›ผ๐‘˜=11โˆ’๐œ†(๐œˆ(๐œŽ๐‘š(๐‘Ž)))๐›ผ,(4.24) we can observe the recurrence ๎‚Š๐‘€det๐‘š(๐œŽ๐‘š1(๐‘Ž))=1โˆ’๐œ†(๐œŽ๐‘š(๐‘Ž))๐›ผ๎‚Š๐‘€det๐‘šโˆ’1๎€ท๐œŽ๐‘šโˆ’1๎€ธ(๐‘Ž),(4.25) which implies the assertion.

Now we summarize the results of Theorem 3.9, Corollary 4.5, and Lemma 4.6 to obtain

Theorem 4.7. Let ๐‘ฆ(๐‘ก) be any solution of (4.1) defined on ๎‚๐•‹[๐œŽ(๐‘Ž),๐‘]โˆฉ๐œŽ(๐‘Ž)(๐‘ž,โ„Ž), where ๎‚๐•‹๐‘โˆˆ๐œŽ(๐‘Ž)(๐‘ž,โ„Ž) is satisfying |๐œ†|(๐œˆ(๐‘))๐›ผ<1. Then ๐‘ฆ(๐‘ก)=๐‘š๎“๐‘—=1๐‘๐‘—๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผโˆ’๐‘š+๐‘—(๐‘ก),(4.26) where ๐‘1,โ€ฆ,๐‘๐‘š are real constants.

We conclude this paper by the illustrating example.

Example 4.8. Consider the initial value problem ๐‘Žโˆ‡๐›ผ(๐‘ž,โ„Ž)๐‘ฆ(๐‘ก)=๐œ†๐‘ฆ(๐‘ก),๐œŽ3(๐‘Ž)โ‰ค๐‘กโ‰ค๐œŽ๐‘›(๐‘Ž),1<๐›ผโ‰ค2,๐‘Žโˆ‡๐›ผโˆ’1(๐‘ž,โ„Ž)||๐‘ฆ(๐‘ก)๐‘ก=๐œŽ2(๐‘Ž)=๐‘ฆ๐›ผโˆ’1,๐‘Žโˆ‡๐›ผโˆ’2(๐‘ž,โ„Ž)||๐‘ฆ(๐‘ก)๐‘ก=๐œŽ2(๐‘Ž)=๐‘ฆ๐›ผโˆ’2,(4.27) where ๐‘› is a positive integer given by the condition |๐œ†|๐œˆ(๐œŽ๐‘›(๐‘Ž))๐›ผ<1. By Theorem 4.7, its solution can be expressed as a linear combination ๐‘ฆ(๐‘ก)=๐‘1๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผโˆ’1(๐‘ก)+๐‘2๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผ(๐‘ก).(4.28) The constants ๐‘1, ๐‘2 can be determined from the system ๐‘‰2โˆ’๐›ผ๎‚€๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผโˆ’1,๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผ๎‚๎€ท๐œŽ2๎€ธโ‹…๎‚ต๐‘(๐‘Ž)1๐‘2๎‚ถ=๎‚ต๐‘ฆ๐›ผโˆ’2๐‘ฆ๐›ผโˆ’1๎‚ถ(4.29) with the matrix elements ๐‘ฃ11=๐‘ฃ22=[1]๐‘ž+๎€ท[๐›ผ]๐‘žโˆ’[1]๐‘ž๎€ธ๐œ†๐œˆ(๐œŽ(๐‘Ž))๐›ผ(1โˆ’๐œ†๐œˆ(๐œŽ(๐‘Ž))๐›ผ)๎€ท๎€ท๐œŽ1โˆ’๐œ†๐œˆ2๎€ธ(๐‘Ž)๐›ผ๎€ธ,๐‘ฃ12=[2]๐‘ž๎€ท[๐›ผ]๐œˆ(๐œŽ(๐‘Ž))+๐‘žโˆ’[2]๐‘ž๎€ธ๐œ†๐œˆ(๐œŽ(๐‘Ž))๐›ผ+1(1โˆ’๐œ†๐œˆ(๐œŽ(๐‘Ž))๐›ผ)๎€ท๎€ท๐œŽ1โˆ’๐œ†๐œˆ2(๎€ธ๐‘Ž)๐›ผ๎€ธ,๐‘ฃ21=[๐›ผ]๐‘ž๐œ†๐œˆ(๐œŽ(๐‘Ž))๐›ผโˆ’1(1โˆ’๐œ†๐œˆ(๐œŽ(๐‘Ž))๐›ผ)๎€ท๎€ท๐œŽ1โˆ’๐œ†๐œˆ2(๎€ธ๐‘Ž)๐›ผ๎€ธ.(4.30) By Lemma 4.6, the matrix ๐‘‰2โˆ’๐›ผ(๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผโˆ’1,๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผ)(๐œŽ2(๐‘Ž)) has a nonzero determinant, hence applying the Cramer rule, we get ๐‘1=๐‘ฆ๐›ผโˆ’2๐‘ฃ22โˆ’๐‘ฆ๐›ผโˆ’1๐‘ฃ12๐‘Š2โˆ’๐›ผ๎‚€๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผโˆ’1,๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผ๎‚๎€ท๐œŽ2๎€ธ,๐‘(๐‘Ž)2=๐‘ฆ๐›ผโˆ’1๐‘ฃ11โˆ’๐‘ฆ๐›ผโˆ’2๐‘ฃ21๐‘Š2โˆ’๐›ผ๎‚€๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผโˆ’1,๐ธ๐‘Ž,๐œ†๐›ผ,๐›ผ๎‚๎€ท๐œŽ2๎€ธ.(๐‘Ž)(4.31)
Now we make a particular choice of the parameters ๐›ผ, ๐‘Ž, ๐œ†, ๐‘ฆ๐›ผโˆ’1 and ๐‘ฆ๐›ผโˆ’2 and consider the initial value problem in the form 1โˆ‡1.8(๐‘ž,โ„Ž)1๐‘ฆ(๐‘ก)=โˆ’3๐‘ฆ(๐‘ก),๐œŽ3(1)โ‰ค๐‘กโ‰ค๐œŽ๐‘›(1),1โˆ‡0.8(๐‘ž,โ„Ž)|||๐‘ฆ(๐‘ก)๐‘ก=๐œŽ2(1)=โˆ’1,1โˆ‡โˆ’0.2(๐‘ž,โ„Ž)|||๐‘ฆ(๐‘ก)๐‘ก=๐œŽ2(1)=1,(4.32) where ๐‘› is a positive integer satisfying ๐œˆ(๐œŽ๐‘›(1))<35/9. If we take the time scale of integers (the case ๐‘ž=โ„Ž=1), then the solution ๐‘ฆ(๐‘ก) of the corresponding initial value problem takes the form ๐‘ฆ(๐‘ก)=145โˆž๎“๐‘˜=0๎‚€โˆ’13๎‚๐‘˜โˆ๐‘กโˆ’2๐‘—=1(๐‘—+1.8๐‘˜โˆ’0.2)โˆ’2(๐‘กโˆ’2)!15โˆž๎“๐‘˜=0๎‚€โˆ’13๎‚๐‘˜โˆ๐‘กโˆ’2๐‘—=1(๐‘—+1.8๐‘˜+0.8)(๐‘กโˆ’2)!,๐‘ก=2,3,โ€ฆ.(4.33) Similarly we can determine ๐‘ฆ(๐‘ก) for other choices of ๐‘ž and โ„Ž. For comparative reasons, Figure 1 depicts (in addition to the above case ๐‘ž=โ„Ž=1) the solution ๐‘ฆ(๐‘ก) under particular choices ๐‘ž=1.2, โ„Ž=0 (the pure ๐‘ž-calculus), ๐‘ž=1, โ„Ž=0.1 (the pure โ„Ž-calculus) and also the solution of the corresponding continuous (differential) initial value problem.

565067.fig.001
Figure 1: ๐›ผ=1.8,๐‘Ž=1,๐œ†=โˆ’1/3,๐‘ฆ๐›ผโˆ’1=โˆ’1,๐‘ฆ๐›ผโˆ’2=1.

Acknowledgments

The research was supported by the research plan MSM 0021630518 โ€œSimulation modelling of mechatronic systemsโ€ of the Ministry of Education, Youth and Sports of the Czech Republic, by Grant P201/11/0768 of the Czech Grant Agency and by Grant FSI-J-10-55 of the FME, Brno University of Technology.

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