Dual identities in fractional difference calculus within Riemann. Thabet Abdeljawad a b

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1 Dual identities in fractional difference calculus within Riemann Thabet Abdeljawad a b a Department of Mathematics, Çankaya University, 06530, Ankara, Turkey b Department of Mathematics and Physical Sciences arxiv:2.5795v2 [math.ds] 8 Jan 203 Prince Sultan University, P. O. Box 66833, Riyadh 586, Saudi Arabia Abstract. We Investigate two types of dual identities for Riemann fractional sums and differences. The first type relates nabla and delta type fractional sums and differences. The second type represented by the Q-operator relates left and right fractional sums and differences. These dual identities insist that in the definition of right fractional differences we have to use both the nabla and delta operators. The solution representation for higher order Riemann fractional difference equation is obtained as well. Keywords: right (left) delta and nabla fractional sums, right (left) delta and nabla Riemann. Q-operator, dual identity. Introduction During the last two decades, due to its widespread applications in different fields of science and engineering, fractional calculus has attracted the attention of many researchers [9, 0,, 23, 24, 25]. Starting from the idea of discretizing the Cauchy integral formula, Miller and Ross [8] and Gray and Zhang [8] obtained discrete versions of left type fractional integrals and derivatives, called fractional sums and differences. Fifteen years later, several authors started to deal with discrete fractional calculus [, 3, 4, 5, 6, 4, 2, 20, 2, 22], benefiting from the theory of time scales originated by Hilger in 988 (see [5]). In this article, we summarize some of the results mentioned in the above references and add more in right type fractional differences and the higher order. Throughout the article we almost agree with the previously presented definitions except for the definition of right fractional difference. We shall figure out that these definitions seem to be more convenient than the previously presented ones by proving some dual identities. These identities fall into two kinds. The first relates nabla type fractional differences and sums to delta ones. The second kind represented by the Q-operator relates left and right fractional sums and differences. This setting enables us to get identities resembling better the ordinary fractional case. Along with the previously mentioned points we are able to fit a reasonable nabla integration by parts formula which remains in accordance with the one obtained in [2] but different from those obtained in [6] and [4]. The obtained dual identities are also used to obtain a delta integration by parts formula from the nabla one. The solution representation for the higher order Riemann fractional difference equation is obtained as well and thus the result in [27] is generalized. We shall see that the higher order Riemann fractional difference initial value problem of non-integer order needs only one initial condition which is not the case for the higher order difference equation (i.e of positive integer order). The article is organized as follows: The remaining part of this section contains summary to some of the basic notations and definitions in delta and nabla calculus. Section 2 contains the definitions in the frame of delta and nabla fractional sums and differences in the Riemann sense. Moreover, some essential lemmas about the commutativity of the different fractional sum operators with the usual difference operators are established. These lemmas are vital to proceed in the next sections. The third section contains some dual identities relating nabla and delta fractional sums and differences in the left and right cases. Using these dual identities, power formulae for nabla left and right fractional sums and a commutative law for nabla left and right sums are obtained. In Section 4 the integration by parts formula for nabla fractional sums and differences obtained in [27] is used by the help of the dual identities to obtain delta integration by parts formulas. Section 5 is devoted to higher order initial fractional difference value problems into Riemann. Finally, in Section 6 the Q-operator is used

2 to relate left and right fractional sums in the nabla and delta case and hence relates delta and nabla Riemann fractional differences. The Q-dual identities obtained in this section expose the validity of the definition of delta and nabla right fractional differences where the delta and nabla operators are used in each of the definitions of delta and nabla right fractional differences. For a natural number n, the fractional polynomial is defined by, n t (n) = (t j) = Γ(t+) Γ(t+ n), () j=0 where Γ denotes the special gamma function and the product is zero when t+ j = 0 for some j. More generally, for arbitrary α, define t (α) = Γ(t+) Γ(t+ α), (2) where the convention that division at pole yields zero. Given that the forward and backward difference operators are defined by f(t) = f(t+) f(t), f(t) = f(t) f(t ) (3) respectively, we define iteratively the operators m = ( m ) and m = ( m ), where m is a natural number. Here are some properties of the factorial function. Lemma.. ([3]) Assume the following factorial functions are well defined. (i) t (α) = αt (α ). (ii) (t µ)t (µ) = t (µ+), where µ R. (iii) µ (µ) = Γ(µ+). (iv) If t r, then t (α) r (α) for any α > r. (v) If 0 < α <, then t (αν) (t (ν) ) α. (vi) t (α+β) = (t β) (α) t (β). Also, for our purposes we list down the following two properties, the proofs of which are straightforward. s(s t) (α ) = (α )(ρ(s) t) (α 2). (4) t(ρ(s) t) (α ) = (α )(ρ(s) t) (α 2). (5) For the sake of the nabla fractional calculus we have the following definition Definition.. ([5, 6, 7, 9]) (i) For a natural number m, the m rising (ascending) factorial of t is defined by (ii) For any real number the α rising function is defined by m t m = (t+k), t 0 =. (6) t α = Γ(t+α), t R {..., 2,,0}, 0 α = 0 (7) Γ(t) Regarding the rising factorial function we observe the following: (i) (t α ) = αt α (8) (ii) (t α ) = (t+α ) (α). (9) (iii) t(s ρ(t)) α = α(s ρ(t)) α (0) Notation: (i) For a real α > 0, we set n = [α]+, where [α] is the greatest integer less than α. (ii) For real numbers a and b, we denote N a = {a,a+,...} and b N = {b,b,...}. (iii) For n N and real a, we denote (iv) For n N and real b, we denote n f(t) ( ) n n f(t). n f(t) ( ) n n f(t). 2

3 2 Definitions and essential lemmas Definition 2.. Let σ(t) = t+ and ρ(t) = t be the forward and backward jumping operators, respectively. Then (i) The (delta) left fractional sum of order α > 0 (starting from a) is defined by: α a f(t) = t α (t σ(s)) (α ) f(s), t N a+α. () s=a (ii) The (delta) right fractional sum of order α > 0 (ending at b) is defined by: b α f(t) = b (s σ(t)) (α ) f(s) = b (ρ(s) t) (α ) f(s), t b α N. (2) +α +α (iii) The (nabla) left fractional sum of order α > 0 (starting from a) is defined by: a f(t) = t s=a+ (t ρ(s)) α f(s), t N a+. (3) (iv)the (nabla) right fractional sum of order α > 0 (ending at b) is defined by: b f(t) = b (s ρ(t)) α f(s) = Regarding the delta left fractional sum we observe the following: (i) α a maps functions defined on N a to functions defined on N a+α. (ii) u(t) = n a f(t), n N, satisfies the initial value problem b (σ(s) t) α f(s), t b N. (4) n u(t) = f(t), t N a, u(a+j ) = 0, j =,2,...,n. (5) (iii) The Cauchy function (t σ(s))(n ) (n )! vanishes at s = t (n ),...,t. Regarding the delta right fractional sum we observe the following: (i) b α maps functions defined on b N to functions defined on b α N. (ii) u(t) = b n f(t), n N, satisfies the initial value problem n u(t) = f(t), t bn, u(b j +) = 0, j =,2,...,n. (6) (iii) the Cauchy function (ρ(s) t)(n ) (n )! vanishes at s = t+,t+2,...,t+(n ). Regarding the nabla left fractional sum we observe the following: (i) a maps functions defined on N a to functions defined on N a. (ii) n a f(t) satisfies the n-th order discrete initial value problem n y(t) = f(t), i y(a) = 0, i = 0,,...,n (7) (iii) The Cauchy function (t ρ(s))n Γ(n) satisfies n y(t) = 0. Regarding the nabla right fractional sum we observe the following: (i) b maps functions defined on b N to functions defined on b N. (ii) b n f(t) satisfies the n-th order discrete initial value problem n y(t) = f(t), i y(b) = 0, i = 0,,...,n. (8) The proof can be done inductively. Namely, assuming it is true for n, we have By the help of (0), it follows that n+ b (n+) f(t) = n [ b (n+) f(t)]. (9) n+ b (n+) f(t) = n b n f(t) = f(t). (20) The other part is clear by using the convention that s k=s+ = 0. (iii) The Cauchy function (s ρ(t))n Γ(n) satisfies n y(t) = 0. 3

4 Definition 2.2. (i)[8] The (delta) left fractional difference of order α > 0 (starting from a ) is defined by: α a f(t) = n (n α) n t (n α) a f(t) = (t σ(s)) (n α ) f(s), t N a+(n α) (2) s=a (ii) [2] The (delta) right fractional difference of order α > 0 (ending at b ) is defined by: b α f(t) = n b (n α) f(t) = ( )n n b (s σ(t)) (n α ) f(s), t b (n α) N (22) +(n α) (iii) The (nabla) left fractional difference of order α > 0 (starting from a ) is defined by: α af(t) = n (n α) n t a f(t) = (t ρ(s)) n α f(s), t N a+ (23) s=a+ (iv) The (nabla) right fractional difference of order α > 0 (ending at b ) is defined by: b α f(t) = n b (n α) f(t) = ( )n n b (s ρ(t)) n α f(s), t b N (24) Regarding the domains of the fractional type differences we observe: (i) The delta left fractional difference α a maps functions defined on Na to functions defined on N a+(n α). (ii) The delta right fractional difference b α maps functions defined on b N to functions defined on b (n α) N. (iii) The nabla left fractional difference α a maps functions defined on N a to functions defined on N a+n. (iv) The nabla right fractional difference b α maps functions defined on b N to functions defined on b n N. Lemma 2.. [3] For any α > 0, the following equality holds: α a f(t) = α a f(t) (t a)α f(a). Lemma 2.2. [2] For any α > 0, the following equality holds: b α f(t) = b α f(t) (b t)α f(b). Lemma 2.3. [7] For any α > 0, the following equality holds: (t a+)α a+ f(t) = α a f(t) f(a) (25) The result of Lemma 2.3 was obtained in [7] by applying the nabla left fractional sum starting from a not from a+. Next will provide the version of Lemma 2.3 by applying the definition in this article. Actually, the nabla fractional sums defined in this article and those in [7] are related. For more details we refer to [27]. Lemma 2.4. For any α > 0, the following equality holds: Proof. By the help of the following by parts identity a f(t) = a f(t) (t a)α f(a). (26) we have s[(t s) α f(s)] = s(t s) α f(s)+(t ρ(s)) α sf(s) (27) = (α )(t ρ(s)) α 2 f(s)+(t ρ(s)) α sf(s) a f(t) = t s=a+ (t ρ(s)) α sf(s) = 4

5 On the other hand [(t s)α f(s) t a (t a)α f(a)+ t s=a+ +(α ) Γ(α ) t (t ρ(s)) α 2 f(s)] = (28) s=a+ t (t ρ(s)) α 2 f(s) (29) s=a+ a f(t) = t(t ρ(s)) α f(s) = Γ(α ) t s=a+ (t ρ(s)) α 2 f(s) (30) Remark 2.. Let α > 0 and n = [α]+. Then, by the help of Lemma 2.4 we have or Then, using the identity α a f(t) = n ( (n α) a f(t)) = n ( (n α) a f(t)). (3) α a f(t) = n [ (n α) a n(t a)n α we infer that (26) is valid for any real α. f(t)+ (t a)n α f(a)] (32) = (t a) α Γ( α) By the help of Lemma 2.4, Remark 2. and the identity (t a) α = (α )(t a) α 2, we arrive inductively at the following generalization. Theorem 2.5. For any real number α and any positive integer p, the following equality holds: a (33) p p f(t) = p a f(t) (t a) α p+k Γ(α+k p+) k f(a). (34) where f is defined on N a and some points before a. Lemma 2.6. For any α > 0, the following equality holds: b f(t) = b f(t) Proof. By the help of the following discrete by parts formula: we have s[(ρ(s) ρ(t)) α f(s)] = (b t)α f(b) (35) (α )(s ρ(t)) α 2 f(s)+(s ρ(t)) α f(s) (36) b a f(t) = b (s ρ(t)) α f(s) = b [ b s((ρ(s) ρ(t)) α f(s))+(α ) (s ρ(t)) α 2 f(s)] = (37) b (s ρ(t)) α 2 f(s) (b t)α f(b). (38) Γ(α ) On the other hand, b f(t) = b where the identity t(s ρ(t)) α f(s) = and the convention that (0) α = 0 are used. t(s ρ(t)) α = (α )(s ρ(t)) α 2 b (s ρ(t)) α 2 f(s) (39) Γ(α ) 5

6 Remark 2.2. Let α > 0 and n = [α]+. Then, by the help of Lemma 2.6 we can have or a b α f(t) = a n ( b (n α) f(t)) = n ( b (n α) f(t)) (40) b α f(t) = n [ b (n α) f(t)+ (b t)n α f(b)] (4) Then, using the identity we infer that (35) is valid for any real α. n(b t)n α = (b t) α Γ(n α) Γ( α) By the help of Lemma 2.6, Remark 2.2 and the identity (b t) α = (α )(b t) α 2, if we follow inductively we arrive at the following generalization Theorem 2.7. For any real number α and any positive integer p, the following equality holds: p b p f(t) = p b (b t) α p+k f(t) Γ(α+k p+) k f(b) (43) where f is defined on b N and some points after b. The following theorem modifies Theorem 2.5 when f is only defined at N a. Theorem 2.8. For any real number α and any positive integer p, the following equality holds: p a+p p f(t) = p a+p f(t) (t (a+p )) α p+k k f(a+p ). (44) Γ(α+k p+) where f is defined on only N a. The proof follows by applying Remark 2. inductively. Similarly, in the right case we have Theorem 2.9. For any real number α and any positive integer p, the following equality holds: (42) p b p+ p f(t) = p b p+ f(t) where f is defined on b N only. (b p+ t) α p+k Γ(α +k p+) k f(b p+) (45) 3 Dual identities for right fractional sums and differences The dual relations for left fractional sums and differences were investigated in [5]. Indeed, the following two lemmas are dual relations between the delta left fractional sums (differences) and the nabla left fractional sums (differences). Lemma 3.. [5] Let 0 n < α n and let y(t) be defined on N a. Then the following statements are valid. (i)( α a )y(t α) = α a y(t) for t N n+a. (ii) ( α a )y(t +α) = a y(t) for t N a. Lemma 3.2. [5] Let 0 n < α n and let y(t) be defined on N α n. Then the following statements are valid. (i) α α n y(t) = ( α α n y)(t +α) for t N n. (ii) (n α) α n y(t) = ( (n α) α n y)(t n+α) for t N 0. 6

7 We remind that the above two dual lemmas for left fractional sums and differences were obtained when the nabla left fractional sum was defined by a f(t) = t (t ρ(s)) α f(s), t N a (46) s=a Now, in analogous to Lemma 3. and Lemma 3.2, for the right fractional summations and differences we obtain Lemma 3.3. Let y(t) be defined on b+ N. Then the following statements are valid. (i)( b α )y(t+α) = b+ α y(t) for t b n N. (ii) ( b α )y(t α) = b+ y(t) for t b N. Proof. We prove only (i). The proof of (ii) is similar and easier. ( b α )y(t+α) = ( ) n n b (n α) y(t+α) = ( ) n n b (s t α) (n α ) y(s) = ( )n n b (s t +n α) (n α ) y(s) (47) +n Using the identity t α = (t+α ) (α), we arrive at ( b α )y(t+α) = ( )n n b (s ρ(t)) n α y(s) = ( ) n n b+ (n α) y(t) = b+ α y(t). (48) Lemma 3.4. Let 0 n < α n and let y(t) be defined on n α N. Then the following statements are valid. (i) n α α y(t) = n α+ α y(t α), t nn (ii) n α (n α) y(t) = n α+ (n α) y(t+n α), t 0 N Proof. We prove (i), the proof of (ii) is similar. By the definition of right nabla difference we have a n n α+ α y(t α) = a n n α α By using (9) it follows that (s ρ(t α)) n α y(s) = n b n α+ α y(t α) = n b n α +n α n α α (s ρ(t α)) n α y(s) = n α +n α (s ρ(t+n α)) n α y(s) (49) (s σ(t)) (n α ) y(s) = n α α y(t) (50) Note that the above two dual lemmas for right fractional differences can not be obtained if we apply the definition of the delta right fractional difference introduced in [4] and [6]. Lemma 3.5. [2] Let α > 0, µ > 0. Then, b µ α (b t) (µ) = Γ(µ+) Γ(µ+α+) (b t)(µ+α) (5) The following commutative property for delta right fractional sums is Theorem 9 in [2]. 7

8 Theorem 3.6. Let α > 0, µ > 0. Then, for all t such that t b (µ+α) (mod ), we have b α [ b µ f(t)] = b (µ+α) f(t) = b µ [ b α f(t)] (52) where f is defined on b N. Proposition 3.7. Let f be a real valued function defined on b N, and let α,β > 0. Then b [ b β f(t)] = b (α+β) f(t) = b β [ b f(t)] (53) Proof. The proof follows by applying Lemma 3.3(ii) and Theorem 3.6 above. Indeed, b [ b β f(t)] = b b β f(t β) = b α b β f(t (α+β)) = b (α+β) f(t (α +µ)) = b (α+β) y(t) (54) The following power rule for nabla right fractional differences plays an important rule. Proposition 3.8. Let α > 0, µ >. Then, for t b N, we have Proof. By the dual formula (ii) of Lemma 3.3, we have b (b t) µ = Γ(µ+) Γ(α+µ+) (b t)α+µ (55) b (b t) µ = b α (b r) µ r=t α = b (s t+α ) (α ) (b s) µ. (56) Then by the identity t α = (t+α ) (α ) and using the change of variable r = s µ+, it follows that b (b t) µ = b µ r=t µ+ Which by Lemma 3.5 leads to (r σ(t α µ+)) (α ) (b r) µ = ( b µ α (b u) µ ) u= α µ++t. (57) b (b t) µ = Γ(µ+) Γ(α+µ+) (b t+α+µ )(α+µ) = Γ(µ+) Γ(α +µ+) (b t)α+µ (58) Similarly, for the nabla left fractional sum we can have the following power formula and exponent law. Proposition 3.9. Let α > 0, µ >. Then, for t N a, we have a (t a)µ = Γ(µ+) Γ(α +µ+) (t a)α+µ (59) Proposition 3.0. Let f be a real valued function defined on N a, and let α,β > 0. Then a [ β a f(t)] = (α+β) a f(t) = β a [ a f(t)] (60) Proof. The proof can be achieved as in Theorem 2. [5], by expressing the left hand side of (60), interchanging the order of summation and using the power formula (59). Alternatively, the proof can be done by following as in the proof of Proposition 3.7 with the help of the dual formula for left fractional sum in Lemma 3. after its arrangement according to our definitions. 8

9 4 Integration by parts for fractional sums and differences In this section we state the integration by parts formulas for nabla fractional sums and differences obtained in [27], then use the dual identities to obtain delta integration by part formulas. Proposition 4.. [27] For α > 0, a,b R, f defined on N a and g defined on b N, we have b s=a+ g(s) a f(s) = b s=a+ Proof. By the definition of the nabla left fractional sum we have b s=a+ g(s) a f(s) = b s=a+ g(s) If we interchange the order of summation we reach at ( 6). f(s) b g(s). (6) s r=a+ (s ρ(r)) α f(r). (62) By the help of Theorem 2.5, Proposition 3.0, (7) and that (n α) a f(a) = 0, the authors in [27] obtained the following left important tools which lead to a nabla integration by parts formula for fractional differences. Proposition 4.2. [27] For α > 0, and f defined in a suitable domain N a, we have α a α a f(t) = f(t), (63) and a α af(t) = f(t), when α / N, (64) n a α a f(t) = f(t) (t a) k k f(a),,when α = n N. (65) k! By the help of Theorem 2.7, Proposition 3.7, (8) and that b (n α) f(b) = 0, the authors also in [27] also obtained the following right important tool: Proposition 4.3. [27] For α > 0, and f defined in a suitable domain b N, we have b α b f(t) = f(t), (66) and b b α f(t) = f(t), when α / N, (67) n b b α f(t) = f(t) (b t) k k! k f(b),when α = n N. (68) Proposition 4.4. [27] Let α > 0 be non-integer and a,b R such that a < b and b a (mod ).If f is defined on b N and g is defined on N a, then b s=a+ f(s) α ag(s) = b s=a+ g(s) b α f(s). (69) The proof was achieved by making use of Proposition 4. and the tools Proposition 4.2 and Proposition 4.3. Now by the above nabla integration by parts formulas and the dual identities in Lemma 3. adjusted to our definitions and Lemma 3.3 we can obtain delta integration by parts formulas. Proposition 4.5. Let α > 0, a,b R such that a < b and b a (mod ). If f is defined on N a and g is defined on b N, then we have b s=a+ b g(s)( α a+ f)(s+α) = s=a+ f(s) b α g(s α). (70) 9

10 Proposition 4.6. Let α > 0 be non-integer and assume that b a (mod ). If f is defined on b N and g is defined on N a, then b s=a+ f(s) α a+g(s α) = b s=a+ g(s) b α f(s+α). (7) 5 Higher order fractional difference initial value problem within Riemann Let α > 0 be a non-integer, n = [α]+ and a(α) = a+n. Consider the fractional initial difference equation α a(α) y(t) = f(t,y(t)), t = a(α) +,a(α)+2,... (72) (n α) a(α) y(t) t=a(α) = y(a(α)) = c. Apply the sum operator to both sides of (72) to get a(α) a(α) { α a(α) Applying (64), we reach at y(t)+ a(α) y(t) + n(t a(α)+)n α +)n α n(t a(α) y(a(α))} = f(t,y(t)). (73) a(α) y(a(α)) = f(t,y(t)). (74) a(α) Now, set g α(t) = (t a(α)+)n α y(a(α)), then by Theorem 2.8 with p = n and f(t) = g α(t) we have n a(α) n g α(t) = n a(α) gα(t) (t a(α)) α n+k Γ(α +k n+) k g α(a(α)) (75) Noting that k g α(a(α)) = y(a(α)) for k = 0,,...,n, by the power formula (59) we conclude that a(α) n g α(t) = 0 n(t a(α)+)α y(a(α)) n Then the substitution in (74) will lead to the following solution representation y(t) = n(t a(α)+)α y(a(α)) As a particular case, if 0 < α < then a(α) = a and hence which is the result obtained in [27]. y(t) = (t a+)α (t a(α)) α n+k y(a(α)). (76) Γ(α+k n+) n (t a(α)) α n+k + Γ(α+k n+) c+ α f(t,y(t)). (77) a(α) y(a)+ a f(t,y(t)), (78) If < α < 2, then n = 2 and a(α) = a+ and the initial value problem (72) becomes From (77), the solution is y(t) = (t a)α 3 c Γ(α 2) Combining the first two terms then α ay(t) = f(t,y(t)), t = a+2,a+3,... (79) (2 α) a y(t) t=a+ = y(a+) = c. + (t a )α 2 c Γ(α ) + (t a )α c + a+f(t,y(t)). (80) 0

11 y(t) = (t a)α 2 c + (t a )α c + a+f(t,y(t)). (8) Γ(α ) Again combining we reach at y(t) = (t a)α c If we proceed inductively we can state + a+f(t,y(t)). (82) Theorem 5.. Let α > 0 be a non-integer, n = [α]+ and a(α) = a+n. Then the solution of the fractional initial difference equation α a(α) y(t) = f(t,y(t)), t = a(α) +,a(α)+2,... (83) (n α) a(α) y(t) t=a(α) = y(a(α)) = c, is given by y(t) = (t a(α) +)α c + f(t,y(t)). (84) a(α) The surprise in Theorem 5. is that the higher order Riemann fractional difference initial value problem of non-integer order needs only one initial condition which is not the case for the higher order difference equation (i.e of positive integer order). Also the solution is consisting of two terms which is not the case for fractional Cauchy problems into Riemann. 6 The Q-operator and fractional difference equations If f(s) is defined on N a b N and a b (mod ) then (Qf)(s) = f(a + b s). The Q-operator generates a dual identity by which the left type and the right type fractional sums and differences are related. Using the change of variable u = a+b s, in [] it was shown that and hence The proof of (86) follows by the definition, (85) and by noting that Similarly, in the nabla case we have and hence α a Qf(t) = Q b α f(t), (85) α a Qf(t) = (Q b α f)(t). (86) Q f(t) = Qf(t). The proof of (88) follows by the definition, (87) and that a Qf(t) = Q b f(t), (87) α a Qf(t) = (Q b α f)(t). (88) Q f(t) = Qf(t). It is remarkable to mention that the Q-dual identity (86) can not be obtained if the definition of the delta right fractional difference introduced by Nuno R.O. Bastos et al. in [4] or by Atıcı F. et al. in [6] is used. Thus, the definition introduced in [] and [2] is more convenience. Analogously, the Q-dual identity (88) indicates that the nabla right Riemann fractional differences presented in this article are also more convenient. It is clear from the above argument that, the Q-operator agrees with its continuous counterpart when applied to left and right fractional Riemann Integrals Riemann derivatives. More generally, this discrete version of the Q-operator can be used to transform the discrete delay-type fractional functional difference dynamic equations to advanced ones. For details in the continuous counterparts see [2].

12 References [] T. Abdeljawad, On Riemann and Caputo fractional differences, Computers and Mathematics with Applications, Volume 62, Issue 3, August 20, Pages [2] T. Abdeljawad (Maraaba), Baleanu D. and Jarad F., Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, Journal of Mathematical Physics, 49 (2008), [3] F.M. Atıcı and Eloe P. W., A Transform method in discrete fractional calculus, International Journal of Difference Equations, vol 2, no 2, (2007), [4] F.M. Atıcı and Eloe P. W., Initial value problems in discrete fractional calculus, Proceedings of the American Mathematical Society, 37, (2009), [5] F. M. Atıcı and Paul W.Eloe, Discrete fractional calculus with the nabla operator, Electronic Journal of Qualitative Theory of Differential equations, Spec. Ed. I, 2009 No.3, 2. [6] F.M. Atıcı, Şengül S., Modelling with fractional difference equations,journal of Mathematical Analysis and Applications, 369 (200) -9. [7] F. M. Atıcı, Paul W. Eloe, Gronwall s inequality on discrete fractional calculus, Copmuterand Mathematics with pplications, In Press, doi:0.06/camwa [8] K. S. Miller, Ross B.,Fractional difference calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, (989), [9] I. Podlubny, Fractional Differential Equations, Academic Press: San Diego CA, (999). [0] Samko G. Kilbas A. A., Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 993. [] Kilbas A., Srivastava M. H.,and Trujillo J. J., Theory and Application of Fractional Differential Equations, North Holland Mathematics Studies 204, [2] T. Abdeljawad, D. Baleanu, Fractional Differences and integration by parts, Journal of Computational Analysis and Applications vol 3 no. 3, (20). [3] Silva M. F., J. A. T. Machado, A. M. Lopes, Modelling and simulation of artificial locomotion systems, Robotica 23 (2005), [4] Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres, Discrete-time fractional variational problems, Signal Processing, 9(3): (20). [5] Bohner M. and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, [6] G. Boros and V. Moll, Iresistible Integrals; Symbols,Analysis and Expreiments in the Evaluation of Integrals, Cambridge University PressCambridge [7] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, A Foundation for Copmuter Science, 2nd ed., Adison-Wesley,Reading, MA, 994. [8] H. L. Gray and N. F.Zhang, On a new definition ofthe fractional difference, Mathematicsof Computaion 50, (82), (988.) [9] J. Spanier and K. B. Oldham, The Pochhammer Polynomials (x) n, An Atlas of Functions, pp ,Hemisphere, Washington, DC, 987. [20] G. A. Anastassiou,Principles of delta fractional calculus on time scales and inequalities, Mathematical and Computer Modelling, 52 (200) [2] G. A. Anastassiou, Nabla discrete calcilus and nabla inequalities, Mathematical and Computer Modelling, 5 (200) [22] G. A. Anastassiou,Foundations of nabla fractional calculus on time scales and inequalities,computer and Mathematics with Applications, 59 (200) [23] O. P. Agrawal, D. Baleanu, A Hamiltonian Formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Contr. 3(9-0) (2007) [24] E. Scalas, Mixtures of compound Poisson processes as models of tick-by-tick financial data. Chaos Solit. Fract. 34() (2007) [25] D. Baleanu, J. J. Trujillo, On exact solutions of a class o f fractional Euler-Lagrange equations, Nonlin.Dyn. 52(4) (2008). [26] T. Maraaba(Abdeljawad), F. Jarad, D. Baleanu, On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Science in China Series A: Mathematics, 5 (0), (2008). [27] T. Abdeljawad, F. Atıcı, On the Definitions of Nabla Fractional Operators, submitted. 2

13 On delta and nabla Caputo fractional differences and dual identities Thabet Abdeljawad a b a Department of Mathematics, Çankaya University, 06530, Ankara, Turkey b Department of Mathematics and Physical Sciences arxiv:2.5795v2 [math.ds] 8 Jan 203 Prince Sultan University, P. O. Box 66833, Riyadh 586, Saudi Arabia Abstract. We Investigate two types of dual identities for Caputo fractional differences. The first type relates nabla and delta type fractional sums and differences. The second type represented by the Q-operator relates left and right fractional sums and differences. Two types of Caputo fractional differences are introduced, one of them (dual one) is defined so that it obeys the investigated dual identities. The relation between Rieamnn and Caputo fractional differences is investigated and the delta and nabla discrete Mittag-Leffler functions are confirmed by solving Caputo type linear fractional difference equations. A nabla integration by parts formula is obtained for Caputo fractional differences as well. Keywords: right (left) delta and nabla fractional sums, right (left) delta and nabla Riemann and Caputo fractional differences. Q-operator, dual identity. Introduction During the last two decades, due to its widespread applications in different fields of science and engineering, fractional calculus has attracted the attention of many researchers [9, 0,, 23, 24, 25]. Starting from the idea of discretizing the Cauchy integral formula, Miller and Ross [8] and Gray and Zhang [8] obtained discrete versions of left type fractional integrals and derivatives, called fractional sums and differences. After then, several authors started to deal with discrete fractional calculus [, 3, 4, 5, 6, 4, 2, 20, 2, 22, 27, 28], benefiting from time scales calculus originated in 988 (see [5]). In [], the concept of Caputo fractional difference was introduced and investigated. In this article we proceed deeply to investigate Caputo fractional differences under two kinds of dual identities. The first kind relates nabla and delta type Caputo fractional differences and the second one, represented by the Q-operator, relates left and right ones. Arbitrary order Riemann and Caputo fractional differences are related as well. By the help of the previously obtained results in [27] and [28] an integration by parts formula for Caputo fractional differences is originated. The article is organized as follows: The remaining part of this section contains summary to some of the basic notations and definitions in delta and nabla calculus. Section 2 contains the definitions in the frame of delta and nabla fractional sums and differences in the Riemann sense. The third section contains some dual identities relating nabla and delta type fractional sums and differences in Riemann sense as previously investigated in [28]. In Section 4 Caputo fractional differences are given and related to the Riemann ones. In section 5, slightly different modified (dual) Caputo fracctional differences are introduced and investigated under some dual identities. Section 6 is devoted to the integration by parts for delta and nabla Caputo fractional differences. Finally, Section 7 contains Caputo type fractional dynamical equations where a nonhomogeneous nabla Caputo fractional difference equation is solved to obtain nabla discrete versions for Mittag- Leffler functions. For the case α = we obtain the discrete nabla exponential function [5]. In addition to this, the Q-operator is used to relate left and right Caputo fractional differences in the nabla and delta case. The Q-dual identities obtained in this section expose the validity of the definition of delta and nabla right Caputo fractional differences. For a natural number n, the fractional polynomial is defined by, n t (n) = (t j) = Γ(t+) Γ(t+ n), () j=0

14 where Γ denotes the special gamma function and the product is zero when t+ j = 0 for some j. More generally, for arbitrary α, define t (α) = Γ(t+) Γ(t+ α), (2) where the convention that division at pole yields zero. Given that the forward and backward difference operators are defined by f(t) = f(t+) f(t), f(t) = f(t) f(t ) (3) respectively, we define iteratively the operators m = ( m ) and m = ( m ), where m is a natural number. Here are some properties of the factorial function. Lemma.. ([3]) Assume the following factorial functions are well defined. (i) t (α) = αt (α ). (ii) (t µ)t (µ) = t (µ+), where µ R. (iii) µ (µ) = Γ(µ+). (iv) If t r, then t (α) r (α) for any α > r. (v) If 0 < α <, then t (αν) (t (ν) ) α. (vi) t (α+β) = (t β) (α) t (β). Also, for our purposes we list down the following two properties, the proofs of which are straightforward. s(s t) (α ) = (α )(ρ(s) t) (α 2). (4) t(ρ(s) t) (α ) = (α )(ρ(s) t) (α 2). (5) For the sake of the nabla fractional calculus we have the following definition Definition.. ([5, 6, 7, 9]) (i) For a natural number m, the m rising (ascending) factorial of t is defined by (ii) For any real number the α rising function is defined by m t m = (t+k), t 0 =. (6) t α = Γ(t+α), t R {..., 2,,0}, 0 α = 0 (7) Γ(t) Regarding the rising factorial function we observe the following: (i) (t α ) = αt α (8) (ii) (iii) Notation: (t α ) = (t+α ) (α). (9) t(s ρ(t)) α = α(s ρ(t)) α (0) (i) For a real α > 0, we set n = [α]+, where [α] is the greatest integer less than α. (ii) For real numbers a and b, we denote N a = {a,a+,...} and b N = {b,b,...}. (iii) For n N and real a, we denote (iv) For n N and real b, we denote n f(t) ( ) n n f(t). n f(t) ( ) n n f(t). 2

15 2 Definitions and essential lemmas Definition 2.. Let σ(t) = t+ and ρ(t) = t be the forward and backward jumping operators, respectively. Then (i) The (delta) left fractional sum of order α > 0 (starting from a) is defined by: α a f(t) = t α (t σ(s)) (α ) f(s), t N a+α. () s=a (ii) The (delta) right fractional sum of order α > 0 (ending at b) is defined by: b α f(t) = b (s σ(t)) (α ) f(s) = b (ρ(s) t) (α ) f(s), t b α N. (2) +α +α (iii) The (nabla) left fractional sum of order α > 0 (starting from a) is defined by: a f(t) = t s=a+ (t ρ(s)) α f(s), t N a+. (3) (iv)the (nabla) right fractional sum of order α > 0 (ending at b) is defined by: b f(t) = b (s ρ(t)) α f(s) = Regarding the delta left fractional sum we observe the following: (i) α a maps functions defined on N a to functions defined on N a+α. (ii) u(t) = n a f(t), n N, satisfies the initial value problem b (σ(s) t) α f(s), t b N. (4) n u(t) = f(t), t N a, u(a+j ) = 0, j =,2,...,n. (5) (iii) The Cauchy function (t σ(s))(n ) (n )! vanishes at s = t (n ),...,t. Regarding the delta right fractional sum we observe the following: (i) b α maps functions defined on b N to functions defined on b α N. (ii) u(t) = b n f(t), n N, satisfies the initial value problem n u(t) = f(t), t bn, u(b j +) = 0, j =,2,...,n. (6) (iii) the Cauchy function (ρ(s) t)(n ) (n )! vanishes at s = t+,t+2,...,t+(n ). Regarding the nabla left fractional sum we observe the following: (i) a maps functions defined on N a to functions defined on N a. (ii) n a f(t) satisfies the n-th order discrete initial value problem n y(t) = f(t), i y(a) = 0, i = 0,,...,n (7) (iii) The Cauchy function (t ρ(s))n Γ(n) satisfies n y(t) = 0. Regarding the nabla right fractional sum we observe the following: (i) b maps functions defined on b N to functions defined on b N. (ii) b n f(t) satisfies the n-th order discrete initial value problem n y(t) = f(t), i y(b) = 0, i = 0,,...,n. (8) The proof can be done inductively. Namely, assuming it is true for n, we have By the help of (0), it follows that n+ b (n+) f(t) = n [ b (n+) f(t)]. (9) n+ b (n+) f(t) = n b n f(t) = f(t). (20) The other part is clear by using the convention that s k=s+ = 0. (iii) The Cauchy function (s ρ(t))n Γ(n) satisfies n y(t) = 0. 3

16 Definition 2.2. (i)[8] The (delta) left fractional difference of order α > 0 (starting from a ) is defined by: α a f(t) = n (n α) n t (n α) a f(t) = (t σ(s)) (n α ) f(s), t N a+(n α) (2) s=a (ii) [2] The (delta) right fractional difference of order α > 0 (ending at b ) is defined by: b α f(t) = n b (n α) f(t) = ( )n n b (s σ(t)) (n α ) f(s), t b (n α) N (22) +(n α) (iii) The (nabla) left fractional difference of order α > 0 (starting from a ) is defined by: α af(t) = n (n α) n t a f(t) = (t ρ(s)) n α f(s), t N a+ (23) s=a+ (iv) The (nabla) right fractional difference of order α > 0 (ending at b ) is defined by: b α f(t) = n b (n α) f(t) = ( )n n b (s ρ(t)) n α f(s), t b N (24) Regarding the domains of the fractional type differences we observe: (i) The delta left fractional difference α a maps functions defined on Na to functions defined on N a+(n α). (ii) The delta right fractional difference b α maps functions defined on b N to functions defined on b (n α) N. (iii) The nabla left fractional difference α a maps functions defined on N a to functions defined on N a+n (on N a if we think f defined at some points before a). (iv) The nabla right fractional difference b α maps functions defined on b N to functions defined on b n N (on b N if we think f defined at some points after b). Lemma 2.. [3] For any α > 0, the following equality holds: α a (t a)α f(t) = α a f(t) f(a). Lemma 2.2. [2] For any α > 0, the following equality holds: b α f(t) = b α f(t) (b t)α f(b). Lemma 2.3. [7] For any α > 0, the following equality holds: a+ f(t) = α a f(t) (t a+)α f(a) (25) The result of Lemma 2.3 was obtained in [7] by applying the nabla left fractional sum starting from a not from a+. Next will provide the version of Lemma 2.3 by applying the definition in this article. Actually, the nabla fractional sums defined in this article and those in [7] are related. For more details we refer to [27]. Lemma 2.4. (see [27] and [28]) For any α > 0, the following equality holds: a (t a)α f(t) = α a f(t) f(a). (26) Remark 2.. (see [27] and [28]) Let α > 0 and n = [α]+. Then, by the help of Lemma 2.4 we have α af(t) = n ( (n α) a f(t)) = n ( (n α) a f(t)). (27) or α a f(t) = n [ (n α) a f(t)+ (t a)n α f(a)] (28) Then, using the identity n(t a)n α we infer that (26) is valid for any real α. = (t a) α Γ( α) (29) 4

17 By the help of Lemma 2.4, Remark 2. and the identity (t a) α = (α )(t a) α 2, we arrive inductively at the following generalization. Theorem 2.5. (see [27] and [28]) For any real number α and any positive integer p, the following equality holds: p a p f(t) = p a f(t) (t a) α p+k Γ(α+k p+) k f(a). (30) where f is defined on N a and some points before a. Lemma 2.6. (see [27] and [28]) For any α > 0, the following equality holds: b f(t) = b f(t) (b t)α f(b). (3) Remark 2.2. (see [27] and [28]) Let α > 0 and n = [α]+. Then, by the help of Lemma 2.6 we can have or b α f(t) = n ( b (n α) f(t)) = n ( b (n α) f(t)) (32) b α f(t) = n [ b (n α) f(t)+ (b t)n α f(b)] (33) Then, using the identity n(b t)n α = (b t) α Γ(n α) Γ( α) we infer that (3) is valid for any real α. By the help of Lemma 2.6, Remark 2.2 and the identity (b t) α = (α )(b t) α 2, if we follow inductively we arrive at the following generalization. Theorem 2.7. (see [27] and [28]) For any real number α and any positive integer p, the following equality holds: p b p f(t) = p b (b t) α p+k f(t) Γ(α+k p+) k f(b) (35) where f is defined on b N and some points after b. 3 Dual identities for fractional sums and Riemann fractional differences The dual relations for left fractional sums and differences were investigated in [5]. Indeed, the following two lemmas are dual relations between the delta left fractional sums (differences) and the nabla left fractional sums (differences). Lemma 3.. [5] Let 0 n < α n and let y(t) be defined on N a. Then the following statements are valid. (i)( α a )y(t α) = α a y(t) for t N n+a. (ii) ( α a )y(t +α) = a y(t) for t N a. Lemma 3.2. [5] Let 0 n < α n and let y(t) be defined on N α n. Then the following statements are valid. (i) α α n y(t) = ( α α n y)(t +α) for t N n. (ii) (n α) α n y(t) = ( (n α) α n y)(t n+α) for t N 0. We remind that the above two dual lemmas for left fractional sums and differences were obtained when the nabla left fractional sum was defined by a f(t) = t (t ρ(s)) α f(s), t N a (36) s=a Now, in analogous to Lemma 3. and Lemma 3.2, for the right fractional summations and differences the author in [28] obtained: (34) 5

18 Lemma 3.3. Let y(t) be defined on b+ N. Then the following statements are valid. (i)( b α )y(t+α) = b+ α y(t) for t b n N. (ii) ( b α )y(t α) = b+ y(t) for t b N. Lemma 3.4. [28] Let 0 n < α n and let y(t) be defined on n α N. Then the following statements are valid. (i) n α α y(t) = n α+ α y(t α), t nn (ii) n α (n α) y(t) = n α+ (n α) y(t+n α), t 0 N Proof. We prove (i), the proof of (ii) is similar. By the definition of right nabla difference we have a n n α+ α y(t α) = a n n α α By using (9) it follows that (s ρ(t α)) n α y(s) = n b n α+ α y(t α) = n b n α +n α n α α (s ρ(t α)) n α y(s) = n α +n α (s ρ(t+n α)) n α y(s) (37) (s σ(t)) (n α ) y(s) = n α α y(t) (38) Note that the above two dual lemmas for right fractional differences can not be obtained if we apply the definition of the delta right fractional difference introduced in [4] and [6]. Lemma 3.5. [2] Let α > 0, µ > 0. Then, b µ α (b t) (µ) = Γ(µ+) Γ(µ+α+) (b t)(µ+α) (39) The following commutative property for delta right fractional sums is Theorem 9 in [2]. Theorem 3.6. Let α > 0, µ > 0. Then, for all t such that t b (µ+α) (mod ), we have where f is defined on b N. b α [ b µ f(t)] = b (µ+α) f(t) = b µ [ b α f(t)] (40) Proposition 3.7. [28] Let f be a real valued function defined on b N, and let α,β > 0. Then b [ b β f(t)] = b (α+β) f(t) = b β [ b f(t)] (4) Proof. The proof follows by applying Lemma 3.3(ii) and Theorem 3.6 above. Indeed, b [ b β f(t)] = b b β f(t β) = b α b β f(t (α+β)) = b (α+β) f(t (α +µ)) = b (α+β) y(t) (42) The following power rule for nabla right fractional differences plays an important rule. Proposition 3.8. ([27], [28]) Let α > 0, µ >. Then, for t b N, we have b (b t) µ = Γ(µ+) Γ(α+µ+) (b t)α+µ (43) 6

19 Proof. By the dual formula (ii) of Lemma 3.3, we have b (b t) µ = b α (b r) µ r=t α = b (s t+α ) (α ) (b s) µ. (44) Then by the identity t α = (t+α ) (α ) and using the change of variable r = s µ+, it follows that b (b t) µ = b µ r=t µ+ Which by Lemma 3.5 leads to (r σ(t α µ+)) (α ) (b r) µ = ( b µ α (b u) µ ) u= α µ++t. (45) b (b t) µ = Γ(µ+) Γ(α+µ+) (b t+α+µ )(α+µ) = Γ(µ+) Γ(α +µ+) (b t)α+µ (46) Similarly, for the nabla left fractional sum we can have the following power formula and exponent law Proposition 3.9. (see [27] and [28]) Let α > 0, µ >. Then, for t N a, we have a (t a)µ = Γ(µ+) Γ(α +µ+) (t a)α+µ (47) Proposition 3.0. (see [27] and [28]) Let f be a real valued function defined on N a, and let α,β > 0. Then a [ β a f(t)] = (α+β) a f(t) = β a [ a f(t)] (48) Proof. The proof can be achieved as in Theorem 2. [5], by expressing the left hand side of (48), interchanging the order of summation and using the power formula (47). Alternatively, the proof can be done by following as in the proof of Proposition 3.7 with the help of the dual formula for left fractional sum in Lemma 3. after its arrangement according to our definitions. 4 Caputo fractional differences In analogous to the usual fractional calculus we can formulate the following definition Definition 4.. Let α > 0, α / N. Then, (i)[] the delta α order Caputo left fractional difference of a function f defined on N a is defined by t (n α) C α af(t) (n α) a n f(t) = (t σ(s)) (n α ) n sf(s) (49) s=a (ii) [] the delta α order Caputo right fractional difference of a function f defined on b N is defined by C b α f(t) b (n α) n f(t) = where n = [α]+. If α = n N, then b +(n α) C α af(t) n f(t) and C b α f(t) n b f(t) (s σ(t)) (n α ) n f(s) (50) 7

20 (iii) the nabla α order Caputo left fractional difference of a function f defined on N a and some points before a, is defined by t (n α) C α af(t) (n α) a n f(t) = (t ρ(s)) n α n f(s) (5) s=a+ (iv) the nabla α order Caputo right fractional difference of a function f defined on b N and some points after b, is defined by b C b α f(t) b (n α) a n f(t) = (s ρ(t)) n α n f(s) (52) If α = n N, then C α a f(t) n f(t) and C b α f(t) a n f(t) It is clear that C α a maps functions defined on Na to functions defined on N a+(n α), and that C b α maps functions defined on b N to functions defined on b (n α) N. Also, it is clear that the nabla left fractional difference α a maps functions defined on N a to functions defined on N a+ n and the nabla right fractional difference b α maps functions defined on b N to functions defined on b +n N. Riemann and Caputo delta fractional differences are related by the following theorem Theorem 4.. [] For any α > 0, we have n C α af(t) = α (t a) (k α) af(t) Γ(k α+) k f(a) (53) and n C b α f(t) = b α (b t) (k α) f(t) Γ(k α+) k f(b). (54) In particular, when 0 < α <, we have C af(t) = α af(t) (t a)( α) f(a). (55) Γ( α) C b f(t) = b α f(t) (b t)( α) f(b) (56) Γ( α) One can note that the Riemann and Caputo fractional differences, for 0 < α <, coincide when f vanishes at the end points. The following identity is useful to transform delta type Caputo fractional difference equations into fractional summations. Proposition 4.2. [] Assume α > 0 and f is defined on suitable domains N a and b N. Then n α C α a+(n α) a f(t) = f(t) (t a) (k) k f(a) (57) k! and n b (n α) α C (b t) (k) b α f(t) = f(t) k f(b) (58) k! In particular, if 0 < α then α C α a+(n α) a f(t) = f(t) f(a) and b (n α) α C b α f(t) = f(t) f(b). (59) Similar to what we have above, for the nabla fractional differences we obtain Theorem 4.3. For any α > 0, we have and n C α a f(t) = α a f(t) (t a) k α Γ(k α+) k f(a) (60) n C b α f(t) = b α (b t) k α f(t) Γ(k α+) k f(b). (6) 8

21 In particular, when 0 < α <, we have and C α af(t) = α af(t) (t a) α f(a) (62) Γ( α) C b α f(t) = b α f(t) (b t) α f(b) (63) Γ( α) Proof. The proof follows by replacing α by n α and p by n in Theorem 5 and Theorem 2.7, respectively. One can see that the nabla Riemann and Caputo fractional differences, for 0 < α <, coincide when f vanishes at the end points. Proposition 4.4. Assume α > 0 and f is defined on suitable domains N a and b N. Then and a In particular, if 0 < α then a n C α a f(t) = f(t) (t a) k k f(a) (64) k! n b b C α f(t) = f(t) (b t) k k! k f(b). (65) C α a f(t) = f(t) f(a) and b C b α f(t) = f(t) f(b) (66) Proof. The proof of (64) follows by the definition and applying Proposition 3.0 and (89) of Proposition 6.2. The proof of (65) follows by the definition and applying Proposition 3.7 and (92) of Proposition 6.3. Using the definition and Proposition 3.9 and Proposition 3.8, we can find the nabla type Caputo fractional differences for certain power functions. For example, for β > 0 and α 0 we have and However, whereas C α a (t a)β = Γ(β) Γ(β α) (t a)β α (67) C b α (b t) β = Γ(β) Γ(β α) (b t)β α. (68) C α a = C b α = 0 (69) α a = (t a)( α) Γ( α), b α = (b t)( α) Γ( α). (70) In the above formulae (67) and (68), we apply the convention that dividing over a pole leads to zero. Therefore the fractional difference when β = α j, j =,2,...,n is zero. Remark 4.. The results obtained in Theorem 4.and afterward agree with those in the usual continuous case (See [] pages 9,96). 5 A dual nabla Caputo fractional difference In the previous section the nabla Caputo fractional difference is defined under the assumption that f is known before a in the left case and under the assumption that f is known after b in the right case. In this section we define other nabla Caputo fractional differences for which not necessary to request any information about f before a or after b. Since we shall show that these Caputo fractional differences are the dual ones for the delta Caputo fractional differences, we call them dual nabla Caputo fractional differences. 9

22 Definition 5.. Let α > 0, n = [α] +, a(α) = a + n and b(α) = b n +. Then the dual nabla left and right Caputo fractional differences are defined by and respectively. C α a(α) f(t) = (n α) n f(t), t N a(α) a+n (7) b(α) C α f(t) = b(α) (n α) n f(t), t b n N, (72) Notice that the Caputo and the dual Caputo differences coincide when 0 < α and differ for higher order. That is for 0 < α C α a(α) f(t) = C α a f(t) and C b(α) α f(t) = C b α f(t). The following proposition states a dual relation between left delta Caputo fractional differences and left nabla (dual) Caputo fractional differences. Proposition 5.. For α > 0, n = [α]+, a(α) = a+n, we have Proof. For t N a+n, we have ( C α a f)(t α) = ( C α a(α) f)(t), t N a+n. (73) t n ( C α a f)(t α) = (t α σ(s)) (n α ) n f(s) s=a = = = t n (t α σ(s)) (n α ) n f(s+n) s=a t (t ρ(r)+n α 2) (n α ) n f(r) (74) r=a+n t (t ρ(r)) n α n f(r) r=a+n = ( C α a(α) f)(t). Analogously, the following proposition relates right delta Caputo fractional differences and right nabla (dual) Caputo fractional differences. Proposition 5.2. For α > 0, n = [α]+, b(α) = b n+, we have ( C b α f)(t+α) = ( C b(α) α f)(t), t b n N. (75) The following theorem modifies Theorem when f is only defined at N a. Theorem 5.3. For any real number α and any positive integer p, the following equality holds: p a+p p f(t) = p a+p f(t) (t (a+p )) α p+k k f(a+p ). (76) Γ(α+k p+) where f is defined on only N a. The proof follows by applying Remark 2. inductively. Similarly, in the right case we have Theorem 5.4. For any real number α and any positive integer p, the following equality holds: p b p+ p f(t) = p b p+ f(t) where f is defined on b N only. (b p+ t) α p+k Γ(α +k p+) k f(b p+) (77) 0

23 Now by using the modified Theorem 5.3 and Theorem 5.4 we have Theorem 5.5. For any α > 0, we have and n C α a(α) f(t) = α a(α) f(t) n C b(α) α f(t) = b(α) α f(t) (t a(α)) k α Γ(k α+) k f(a(α)) (78) (b(α) t) k α Γ(k α+) k f(b(α)). (79) In particular, when 0 < α <, then a(α) = a and b(α) = b and hence we have and C α a f(t) = α a C b α f(t) = b α f(t) (t a) α f(t) f(a) (80) Γ( α) (b t) α f(b) (8) Γ( α) Also, by using the modified Theorem 5.3 and Theorem 5.4 we have Proposition 5.6. Assume α > 0 and f is defined on suitable domains N a and b N. Then n C α a(α) a(α) f(t) = f(t) (t a(α)) k k f(a(α)) (82) k! and n b(α) C (b(α) t) k b(α) α f(t) = f(t) k f(b(α)). (83) k! In particular, if 0 < α then a(α) = a and b(α) = b and hence a C α a f(t) = f(t) f(a) and b b C α f(t) = f(t) f(b) (84) 6 Integration by parts for Caputo fractional differences In this section we state the integration by parts formulas for nabla fractional sums and differences obtained in [27], then use the dual identities to obtain delta integration by part formulas. Proposition 6.. [27] For α > 0, a,b R, f defined on N a and g defined on b N, we have b s=a+ g(s) a f(s) = b s=a+ f(s) b g(s). (85) Proof. By the definition of the nabla left fractional sum we have b g(s) a f(s) = b s g(s) (s ρ(r)) α f(r). (86) s=a+ s=a+ r=a+ If we interchange the order of summation we reach at ( 85). By the help of Theorem 5, Proposition 3.0, (7) and that (n α) a f(a) = 0, the authors in [27] obtained the following left important tools which lead to a nabla integration by parts formula for fractional differences. Proposition 6.2. [27] For α > 0, and f defined in a suitable domain N a, we have α a a f(t) = f(t), (87) and a α af(t) = f(t), when α / N, (88) n a α a f(t) = f(t) (t a) k k f(a),,when α = n N. (89) k!