Mathematics Subject Classification: 33D60, 33E12, 26A33Based on the fractional q–integral with the parametric lower limit of
integration, we consider the fractional q–derivative of Caputo type. 
Especially, its applications to q-exponential functions allow us to introduce
q–analogues of the Mittag–Leffler function. Vice versa, those functions can
be used for defining generalized operators in fractional q–calculus

Marinkovic, Sladjana

Rajkovic, Predrag

Stankovic, Miomir

Fractional Calculus and Applied Analysis

Bulgarian Digital Mathematics Library at IMI-BAS

ON q–ANALOGUES OF CAPUTO DERIVATIVEAND MITTAG–LEFFLER FUNCTIONPredrag M. Rajkovic´ 1, Sladjana D. Marinkovic´ 2,Miomir S. Stankovic´ 3AbstractBased on the fractional q–integral with the parametric lower limit ofintegration, we consider the fractional q–derivative of Caputo type. Es-pecially, its applications to q-exponential functions allow us to introduceq–analogues of the Mittag–Leﬄer function. Vice versa, those functions canbe used for defining generalized operators in fractional q–calculus.Mathematics Subject Classification: 33D60, 33E12, 26A33Key Words and Phrases: Mittag-Leﬄer function, q–integral, q–derivative,fractional integral, fractional derivative1. IntroductionThe calculus of the real order derivatives and integrals has become verysuitable apparatus in describing and solving a lot of problems in numeroussciences, such as physics, electrochemistry and material science (see, forexample [12]). Their treatment from the point of view of the q–calculus canadditionally open new perspectives as it did, for example, in optimal controlproblems [5].Starting from the q–integral(Iq,0f)(x) =∫ x0f(t) dqt = x(1− q)∞∑k=0f(xqk) qk (0 ≤ |q| < 1) , (1)360 P.M. Rajkovic´, S.D. Marinkovic´, M.S. Stankovic´and (Iq,af)(x) =∫ xaf(t) dqt =∫ x0f(t) dqt−∫ a0f(t) dqt , (2)the iterated q–integral operator Inq,a is defined byI0q,af = f, Inq,af = Iq,a(In−1q,a f)(n = 1, 2, 3, . . .) .The reduction of this iterated q–integral to a single integral of one variablewas considered by Al-Salam [3], by a q–analogue of the Cauchy formula(Inq,af)(x) =xn−1[n− 1]q!∫ xa(qt/x; q)n−1 f(t) dqt (n ∈ N) , (3)where[a]q :=1− qa1− q (a ∈ R) , [n]q! =n∏k=1[k]q! (n ∈ N),and(a; q)∞ =∞∏i=0(1− aqi) , (a; q)α = (a; q)∞(aqα; q)∞ (a, α ∈ R) . (4)Al-Salam [2] and Agarwal [1] introduced several types of fractional q–integral operators and fractional q–derivatives, always with the lower limitof integration being zero. However, in some considerations, such as theconstruction of a q–Taylor formula or solving of q–differential equation offractional order, it is interesting to allow nonzero lower limit of integration.Therefore, we define the fractional q–integral in the following way.Definition 1.1. The fractional q–integral is(Iαq,cf)(x) =xα−1Γq(α)∫ xc(qt/x; q)α−1 f(t) dqt (α ∈ R+) , (5)where the q–Gamma function is defined byΓq(x) = (q; q)x−1(1− q)1−x . (6)Lemma 1.1. The fractional q–integral (5) can be written in the equiv-alent form (Iαq,cf)(x) =∫ xcf(t) dqwα(x, t) (α ∈ R+) ,ON q–ANALOGUES OF CAPUTO DERIVATIVE. . . 361where the function wα(x, t) is defined bywα(x, t) =1Γq(α+ 1)(xα − xα(t/x; q)α)(α ∈ R+) .The permission for the lower limit of integration to take some nonzerovalue, brings a lot of troubles while working with fractional q–calculus (see[13]).Here are some of the properties of the previously defined integral.Theorem 1.2. Let α, β ∈ R+. The q–fractional integration has thefollowing semigroup property(Iβq,cIαq,cf)(x) =(Iα+βq,c f)(x) (0 < c < x) .In [13] the next useful statement is proven.Lemma 1.3. For α ∈ R+, λ, λ + α ∈ R \ {−1,−2, . . .}, the followingfractional q–integral is valid:Iαq,c(xλ(c/x; q)λ)=Γq(λ+ 1)Γq(α+ λ+ 1)xα+λ(c/x; q)α+λ (0 < c < x) . (7)2. The fractional q–derivative of Riemann-Liouville typeThe q–derivative of a function f(x) is defined by(Dqf)(x) =f(x)− f(qx)x− qx (x 6= 0) ,(Dqf)(0) = limx→0(Dqf)(x) ,and the q–derivatives of higher order, resp.:D0qf = f , Dnq f = Dq(Dn−1q f)(n = 1, 2, 3, . . .) . (8)On the basis of fractional q–integral, we can define q–derivative of realorder.Definition 2.1. The fractional q–derivative of Riemann–Liouvilletype of order α ∈ R+ is(Dαq,cf)(x) =(Ddαeq Idαe−αq,c f)(x), (9)where dαe denotes the smallest integer greater or equal to α.Some useful relations are true:362 P.M. Rajkovic´, S.D. Marinkovic´, M.S. Stankovic´Lemma 2.1. For 0 < c < x, the operators Dαq,c and Iαq,c satisfy:(Dαq,cIαq,cf)(x) = f(x) .Lemma 2.2. For λ ∈ R \ {−1,−2, . . .} and 0 < c < x, the followingrelation is valid:Dαq,c(xλ(c/x; q)λ)=Γq(λ+ 1)Γq(λ− α+ 1) xλ−α(c/x; q)λ−α , α− λ ∈ R \ N ,0 , α− λ ∈ N .3. The fractional q–derivative of Caputo typeIf we change the order of operators in (9), we can introduce anothertype of fractional q–derivative.Definition 3.1. The fractional q-derivative of Caputo type of orderα ∈ R+ is defined as(?Dαq,cf)(x) =(Idαe−αq,c Ddαeq f)(x) . (10)It is important to establish the connection between the two types of thefractional q–derivatives (9) and (10).Theorem 3.1. Let α ∈ R+ \ N0 and 0 < c < x. The following con-nection between the Caputo type and the Riemann-Liouville type fractionalintegral holds true:(Dαq,cf)(x) =(?Dαq,cf)(x) +dαe−1∑k=0(Dkq f)(c)Γq(1 + k − α)xk−α(c/x; q)k−α .The proof is pretty long, so we will omit it here.Lemma 3.2. For α ∈ R+ \ N and λ ∈ (−1,∞), the following isvalid:?Dαq,c(xλ(c/x; q)λ)=Dαq,c(xλ(c/x; q)λ), dαe − λ ∈ R \ N ,0 , dαe − λ ∈ N .ON q–ANALOGUES OF CAPUTO DERIVATIVE. . . 363Theorem 3.3. For α ∈ R+ \ N0 and a < x, we have the followingrelation: (?Dα+1q,a f)(x) =(?Dαq,aDqf)(x).P r o o f. Since α ∈ R+ \N0, we can write α = n+ ε, n ∈ N0, 0 < ε < 1.Then α+ 1 ∈ (n+ 1, n+ 2), so we get(?Dα+1q,a f)(x) =(I1−εq,a Dn+2q f)(x)=(I1−εq,a Dn+1q Dqf)(x) = (?Dαq,aDqf)(x) .Theorem 3.4. For α ∈ R+ \ N0 and a < x,(Dq ?Dαq,af)(x)− (?Dα+1q,a f)(x) = (Ddαeq f)(a)Γq(dαe − α) xdαe−α−1(a/x; q)dαe−α−1 .Theorem 3.5. Let α ∈ R+ \N. Then, for a < x, the following is valid:(Iαq,a ?Dαq,af)(x) = f(x)−dαe−1∑k=0(Dkq f)(a)[k]q!xk(a/x; q)k .P r o o f. Really, we have(Iαq,a ?Dαq,af)(x) =(Iαq,aIdαe−αq,a Ddαeq f)(x) =(Idαeq,a Ddαeq f)(x)= f(x)−dαe−1∑k=0(Dkq f)(a)[k]q!xk(a/x; q)k .Theorem 3.6. Let α ∈ R+ \N. Then, for a < x, the following is valid:(?Dαq,aIαq,af)(x) = f(x) .P r o o f. According to the previous theorems, we can write(?Dαq,aIαq,af)(x) =(Dαq,aIαq,af)(x)−dαe−1∑k=0(Dkq Iαq,af)(a)Γq(1 + k − α)xk−α(a/x; q)k−α= f(x)−dαe−1∑k=0(Iα−kq,a f)(a)Γq(1 + k − α)xk−α(a/x; q)k−α = f(x) .364 P.M. Rajkovic´, S.D. Marinkovic´, M.S. Stankovic´Theorem 3.7. Let α ∈ R+ \ N, β ∈ R+ and α > β. Then, for a < x,the following relations hold:(?Dαq,aIβq,af)(x)=(?Dα−βq,a f)(x) +dα−βe−1∑k=0(Dkq f)(a)Γq(k − α+ β + 1)xk−α+β(a/x; q)k−α+β ,(Iβq,a ?Dαq,af)(x)=(?Dα−βq,a f)(x)−dαe−1∑k=dα−βe(Dkq f)(a)Γq(k − α+ β + 1)xk−α+β(a/x; q)k−α+β .For ideas of the proofs of the propositions in this section, one can see in[13] and [14].4. On q–analogs of the Mittag–Leﬄer functionIn the mathematical literature (see for example, see [12], [8], [9]) theMittag-Leﬄer (M-L) function (of two indices) is already well known and isdefined byEα,β(x) =∞∑n=0znΓ(αn+ β)(α, β ∈ C : Re(α) > 0, Re(β) > 0) .(11)The function Eα(x) = Eα,1(x), as a direct generalization of the exponentialand trigonometric functions, was introduced first in the paper [11] by G.Mittag–Leﬄer in 1903. It always appears when solving fractional orderdifferential or integral equations. Nowadays, it is involved in the treatmentof many concrete problems in various applied sciences.The q–exponential functions (see [6]) can be written by the power serieseq(x) =1(x; q)∞=∞∑n=0xn(q; q)n(|x| < 1) ,Eq(x) = (−x; q)∞ =∞∑n=0q(n2)(q; q)nxn ,or, applying the q–form of the Taylor theorem, byON q–ANALOGUES OF CAPUTO DERIVATIVE. . . 365eq(x) = eq(c)∞∑n=0xn(c/x; q)n(q; q)n(c ∈ R; |c| < |x|) , (12)Eq(x) = Eq(c)∞∑n=0q(n2)(−c; q)nxn(c/x; q)n(q; q)n(c, x ∈ R) . (13)Corollary 4.1. For α ∈ R+ and 0 < c < x, the following q–fractionalintegrals are valid:Iαq,c(eq(x))= (1− q)α eq(c)∞∑n=0xα+n(c/x; q)α+n(q; q)α+n,Iαq,c(Eq(x))= (1− q)α Eq(c)∞∑n=0q(n2)(−c; q)nxα+n(c/x; q)α+n(q; q)α+n.P r o o f. According to Lemma 1.3, we haveIαq,c(xn(c/x; q)n)(q; q)n=1(q; q)nΓq(n+ 1)Γq(n+ α+ 1)xn+α(c/x; q)n+α= (1− q)α xn+α(c/x; q)n+α(q; q)α+n.Applying this to formulas (12) and (13) we get the required identities.The previous corollary indicates that it is necessary to define functionswhich are q–analogues of the Mittag–Leﬄer function (11).Definition 4.1. The functioneq;α,β(x; c) =∞∑n=0xαn+β−1(c/x; q)αn+β−1(q; q)αn+β−1(|c| < |x|) , (14)(q, x, c, α, β ∈ C; Re(α),Re(β) > 0, |q| < 1) (15)we shall call the small q–Mittag–Leﬄer function. Similarly, the big q–Mittag–Leﬄer function is introduced asEq;α,β(x; c) =∞∑n=0q(αn+β−12 )(−c; q)αn+β−1xαn+β−1(c/x; q)αn+β−1(q; q)αn+β−1, (16)with the same conditions (15).366 P.M. Rajkovic´, S.D. Marinkovic´, M.S. Stankovic´Evidently, in the limit case, we getlimq→1limc→0eq;α,β((1− q)x; c) = limq→1limc→0Eq;α,β((1− q)x; c) = xβ−1Eα,β(xα) ,and especially,eq;1,1(x; 0) = eq(x), Eq;1,1(x; 0) = Eq(x) .Theorem 4.2. The following q–fractional integrals are validIαq,c(eq(x))= (1− q)α eq(c) eq;1,α+1(x; c)Iαq,c(Eq(x))= (1− q)αq(α+12 ) Eq(cq−α)Eq;1, α+1(xq−α; cq−α).P r o o f. The first statement is obvious. The second one requires someadditional simplification.Applying the relation (4) and (4), we can write(−c; q)n = (−cq−α; q)n+α(−cq−α; q)α =(−cq−α; q)n+α (−c; q)∞(−cq−α; q)∞= (−cq−α; q)n+α Eq(c)Eq(cq−α).The second statement of Corollary 4.1 can be written in the formIαq,c(Eq(x))= (1− q)α Eq(c)∞∑n=0q(n2)(−c; q)nxα+n (c/x; q)α+n(q; q)α+n= (1− q)α q(α+12 ) Eq(cq−α)×∞∑n=0q(n+α2 )(−cq−α; q)n+α(xq−α)α+n (cq−α/xq−α; q)α+n(q; q)α+n,wherefrom the second relation follows.Corollary 4.3. For α ∈ R\N0, the Riemann-Liouville (R-L) fractionalq–derivatives of the q–exponential functions are:Dαq,c(eq(x)) =eq(c)(1− q)α∞∑n=0xn−α(c/x; q)n−α(q; q)n−αON q–ANALOGUES OF CAPUTO DERIVATIVE. . . 367=eq(c)(1− q)αdαe−1∑n=0xn−α(c/x; q)n−α(q; q)n−α+ eq;1,dαe−α+1(x; c) ,andDαq,c(Eq(x))=Eq(c)(1− q)α∞∑n=0q(n2)(−c; q)nxn−α(c/x; q)n−α(q; q)n−α=q(α2)Eq(cqα)(1− q)αdαe−1∑n=0q(n−α2 )(xqα)n−α(c/x; q)n−α(−cqα; q)n−α (q; q)n−α +Eq;1,dαe−α+1(xqα; cqα) .Corollary 4.4. For α ∈ R\N0, the Caputo (C) fractional q–derivativesof the q–exponential functions are:?Dαq,c(eq(x)) =eq(c)(1− q)α∞∑n=dαexn−α(c/x; q)n−α(q; q)n−α=eq(c)(1− q)α eq;1,dαe−α+1(x; c) ,and?Dαq,c(Eq(x))=Eq(c)(1− q)α∞∑n=dαeq(n2)(−c; q)nxn−α(c/x; q)n−α(q; q)n−α=q(α2)Eq(cqα)(1− q)α Eq;1,dαe−α+1(xqα; cqα) .From now on, we suppose that 0 < c < x and α, β ∈ R.Theorem 4.5. For α, β > 0, the following is true:eq;α,β(x; c) =xβ−1(c/x; q)β−1(q; q)β−1+ eq;α,β+α(x; c) ,Eq;α,β(x; c) =q(β−12 )(−c; q)β−1xβ−1(c/x; q)β−1(q; q)β−1+ Eq;α,β+α(x; c) .P r o o f. Starting from (14), by shifting of the counter index, we haveeq;α,β+α(x; c) =∞∑n=0xα(n+1)+β−1(c/x; q)α(n+1)+β−1(q; q)α(n+1)+β−1=∞∑n=0xαn+β−1(c/x; q)αn+β−1(q; q)αn+β−1− xβ−1(c/x; q)β−1(q; q)β−1.368 P.M. Rajkovic´, S.D. Marinkovic´, M.S. Stankovic´We get the second identity in the same way, starting from (16).Theorem 4.6. For m ∈ N and β > m, the following differentiationformulas hold:Dmq(eq;α,β(x; c))= (1− q)−m eq;α,β−m(x; c) ,Dmq(Eq;α,β(x; c))= (1− q)−m q(m2 )(−c; q)mEq;α,β−m(xqm; cqm) .The fractional q–integrals of the q–Mittag–Leﬄer functions are given inthe next statement.Theorem 4.7. For α, β, µ > 0, the following q–integrals are valid:Iµq,c(eq;α,β(x; c))= (1− q)µ eq;α,β+µ(x; c) ,Iµq,c(Eq;α,β(x; c))= (1− q)µ q(µ+12 )(−cq−µ; q)µ Eq;α,β+µ(xq−µ; cq−µ) .The following two special cases are interesting. The second one followsfrom Theorem 4.5.Corollary 4.8. For α, β > 0, the following formulas hold:Iβq,c(eq;α,β(x; c))= (1− q)β eq;α,2β(x; c) ,Iβq,c(Eq;α,β(x; c))= (1− q)β q(β+12 )(−cq−β; q)β Eq;α,2β(xq−β; cq−β) .Corollary 4.9. For α, β > 0, the following is true:Iαq,c(eq;α,β(x; c))= (1− q)α(eq;α,β(x; c)− xβ−1(c/x; q)β−1(q; q)β−1),Iαq,c(Eq;α,β(x; c))= (1− q)α q(α+12 )(−cq−α; q)α Eq;α,β(xq−α; cq−α)− (1− q)α q(−α+β−12 )(−c; q)−α+β−1xβ−1(c/x; q)β−1(q; q)β−1.Corollary 4.10. For α, β > 0, the following differentiation formulasare true:Dβq,c(eq;α,β(x; c))= (1− q)−β eq;α,α(x; c) ,Dβq,c(Eq;α,β(x; c))= (1− q)−β q(β2)(−c; q)β Eq;α,α(xqβ; cqβ) .ON q–ANALOGUES OF CAPUTO DERIVATIVE. . . 369Corollary 4.11. For α, β > 0, α 6= β, the following is true:Dαq,c(eq;α,β(x; c))= (1− q)−α(eq;α,β(x; c) +xβ−α−1(c/x; q)β−α−1(q; q)β−α−1),Dαq,c(Eq;α,β(x; c))= (1− q)−α q(α2)(−c; q)α×(Eq;α,β(xqα; cqα)− q(β−α−12 )(−cqα; q)β−α−1(xqα)β−α−1(c/x; q)β−α−1(q; q)β−α−1).Theorem 4.12. For α, β > 0, such α−β ∈ N0, the function eq;α,β(τx; τc)(τ ∈ R) is an eigenfunction of the operator Dαq,c, i.e.Dαq,c(eq;α,β(τx; τc))=τα(1− q)α eq;α,β(τx; τc) .P r o o f. Using (14) and Lemma 2.2, and by shifting of the counterindex, we getDαq,c(eq;α,β(τx; τc))=∞∑n=0ταn+β−1 Dαq,c(xαn+β−1(c/x; q)αn+β−1(q; q)αn+β−1)=τα(1− q)α∞∑n=1ταn+β−α−1xαn+β−α−1(c/x; q)αn+β−α−1(q; q)αn+β−α−1=τα(1− q)α∞∑n=0(τx)αn+β−1(c/x; q)αn+β−1(q; q)αn+β−1=τα(1− q)α eq;α,β(τx; τc) .Theorem 4.13. For α, β > 0, such that dαe − β ∈ N0, the functioneq;α,β(τx; τc) (τ ∈ R) is an eigenfunction of the operator ?Dαq,c, i.e.?Dαq,c(eq;α,β(τx; τc))=τα(1− q)α eq;α,β(τx; τc) .P r o o f. By using (14) we get?Dαq,c(eq;α,β(τx; τc))= ?Dαq,c( ∞∑n=0(τx)αn+β−1(c/x; q)αn+β−1(q; q)αn+β−1)=∞∑n=0ταn+β−1 ?Dαq,c(xαn+β−1(c/x; q)αn+β−1(q; q)αn+β−1).370 P.M. Rajkovic´, S.D. Marinkovic´, M.S. Stankovic´If dαe − β = m ∈ N0, according to Lemma 3.2, we have that?Dαq,c(xβ−1(c/x; q)β−1)= 0 .Therefore,?Dαq,c(eq;α,β(τx; τc))=∞∑n=1ταn+β−α−1 ?Dαq,c(xαn+β−1(c/x; q)αn+β−1(q; q)αn+β−1)=τα(1− q)α∞∑n=1ταn+β−α−1xαn+β−α−1(c/x; q)αn+β−α−1(q; q)αn+β−α−1=τα(1− q)α∞∑n=0(τx)αn+β−1(c/x; q)αn+β−1(q; q)αn+β−1=τα(1− q)α eq;α,β(τx; τc) .5. From the q–analogues of the Mittag–Leﬄer functionto q–integrals and q–derivativesFollowing the notions and considerations in Kiryakova [8, Ch.2, Ch.5]and [9], we can define the fractional q–integral and fractional q–derivativeof any function, using the q-analogues of Mittag-Leﬄer functions as gen-erating functions of some Gelfond-Leontiev (G-L) operators for generalizedintegration and differentiation.Let us consider a function ϕ(x; q) of the formϕ(x; q) =∞∑n=0ϕn(q)xαn+β−1(c/x; q)αn+β−1 . (17)By means of its coefficients ϕn(q), we can define two G-L operators: ofintegration LIα,βϕ,q and resp., of differentiation LDα,βϕ,q , in the following way:if a function f(x) is given by the expansionf(x) =∞∑n=0anxαn+β−1(c/x; q)αn+β−1 ,thenLIα,βϕ,q f(x) :=∞∑n=0anϕn+1(q)ϕn(q)xα(n+1)+β−1(c/x; q)α(n+1)+β−1 , (18)ON q–ANALOGUES OF CAPUTO DERIVATIVE. . . 371andLDα,βϕ,q f(x) :=∞∑n=1anϕn−1(q)ϕn(q)xα(n−1)+β−1(c/x; q)α(n−1)+β−1 . (19)The following statement is then true.Theorem 5.1. If ϕ(x; q) = eq;α,1((1 − q)x; (1 − q)c), then LIα,1ϕ,q isa fractional q–integral operator Iαq,c, and LDα,1ϕ,q is a Caputo fractional q–derivative operator ?Dαq,c.P r o o f. The expansion of the function ϕ(x; q) = eq;α,1((1−q)x; (1−q)c)in the form (17) has coefficientsϕn(q) =(1− q)αn(q; q)αn(n ∈ N0) .Hence, forf(x) =∞∑n=0an xαn(c/x; q)αn ,we haveLIα,1ϕ,q f(x) =∞∑n=0an(1− q)α(n+1)(q; q)α(n+1)(q; q)αn(1− q)αn xα(n+1)(c/x; q)α(n+1)= (1− q)α∞∑n=0an(q; q)αn(q; q)α(n+1)xα(n+1)(c/x; q)α(n+1)=∞∑n=0an Iαq,c(xαn(c/x; q)αn)= Iαq,cf(x) .Similarly,LDα,1ϕ,qf(x) =∞∑n=1an(1− q)α(n−1)(q; q)α(n−1)(q; q)αn(1− q)αn xα(n−1)(c/x; q)α(n−1)= (1− q)−α∞∑n=1an(q; q)αn(q; q)α(n−1)xα(n−1)(c/x; q)α(n−1)=∞∑n=1an ?Dαq,c(xαn(c/x; q)αn)= ?Dαq,c( ∞∑n=0anxαn(c/x; q)αn)= ?Dαq,cf(x) .372 P.M. Rajkovic´, S.D. Marinkovic´, M.S. Stankovic´Example 5.1. Especially, if α = 1, the operators LIα,1ϕ,q and LDα,1ϕ,q be-come fractional q–integral operator Iq,c and fractional q–derivative operatorDq, respectively.AcknowledgementsThis work was supported by the Ministry of Science, Technology andDevelopment of Republic Serbia, under Projects No 144023 and No 144013.References[1] R.P. Agarwal, Certain fractional q–integrals and q–derivatives. Proc.Camb. Phil. Soc. 66 (1969), 365-370.[2] W.A. Al-Salam, Some fractional q–integrals and q–derivatives. Proc.Edinburgh Math. Soc. 15 (1966), 135-140.[3] W.A. Al-Salam, q–Analogues of Cauchy’s Formulas. Proc. Amer.Math. Soc. 17, No. 3 (1966), 616-621.[4] W.A. Al-Salam, A. Verma, A fractional Leibniz q–formula. Pacific J.of Mathematics 60, No 2 (1975), 1-9.[5] G. Bangerezako, Variational calculus on q–nonuniform lattices. J.Math. Anal. and Appl. 306, No 1 (2005), 161-179.[6] G. Gasper, M. Rahman, Basic Hypergeometric Series, 2nd Ed, En-cyclopedia of Math. and its Appl. 96. Cambridge University Press,Cambridge (2004).[7] H. Gauchman, Integral inequalities in q–calculus. Computers andMath. with Appl. 47 (2004), 281-300.[8] V. Kiryakova, Generalized Fractional Calculus and Applications. Long-man (Pitman Res. Notes in Math. Ser. 301), Harlow & J. Wiley Ltd.,N. York (1994).[9] V. Kiryakova, Multiple (multiindex) Mittag-Leﬄer functions and re-lations to generalized fractional calculus. J. Comp. Appl. Math. (Spe-cial Issue: ”Higher Transcendental Functions and Their Applications”)118 (2000), 214-259.[10] R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differentialequations of fractional order. In: A. Carpinteri, F. Mainardi (Eds.),Fractals and Fractional Calculus in Continuum Mechanics, Springer,Wien (1997), 223276 (Available at http://www.fracalmo.org).[11] G.M. Mittag-Leﬄer, Sur la nouvelle fonction Eα(x). C. R. Acad. Sci.Paris 137 (1903), 554558.ON q–ANALOGUES OF CAPUTO DERIVATIVE. . . 373[12] I. Podlubny, Fractional Differential Equations (An Introduction toFractional Derivatives, Fractional Differential Equations, Some Meth-ods of Their Solution and Some of Their Applications). AcademicPress, San Diego-Boston-New York-London-Tokyo-Toronto (1999).[13] P.M. Rajkovic´, S.D. Marinkovic´, M.S. Stankovic´, Fractional integralsand derivatives in q–calculus. Applicable Anal. and Discrete Mathe-matics 1 (2007), 311-323.[14] M.S. Stankovic´, P.M. Rajkovic´, S.D. Marinkovic´, On q–fractionalderivatives of Riemann–Liouville and Caputo type. To appear.Received: October 12, 20071 Dept. of Mathematics, Faculty of Mechanical EngineeringUniversity of Niˇs , A. Medvedeva, 1418 000 - Niˇs, SERBIAe-mail: pedja.rajk@yahoo.com2 Dept. of Mathematics, Faculty of Electronic EngineeringUniversity of Niˇs , A. Medvedeva, 1418 000 - Niˇs, SERBIAe-mail: sladjana@elfak.ni.ac.yu3 Dept. of Mathematics, Faculty of Occupational SafetyUniversity of Niˇs, Cˇarnojevic´a 10a,18 000 - Niˇs, SERBIAe-mail: miomir.stankovic@gmail.com

On q–Analogues of Caputo Derivative and Mittag–Leffler Function

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On q–Analogues of Caputo Derivative and Mittag–Leffler Function

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