Research article

On generalized k-fractional derivative operator

  • Received: 11 November 2019 Accepted: 03 February 2020 Published: 19 February 2020
  • MSC : 33C05, 33C15

  • The principal aim of this paper is to introduce k-fractional derivative operator by using the definition of k-beta function. This paper establishes some results related to the newly defined fractional operator such as the Mellin transform and the relations to k-hypergeometric and k-Appell's functions. Also, we investigate the k-fractional derivative of k-Mittag-Leffler and the Wright hypergeometric functions.

    Citation: Gauhar Rahman, Shahid Mubeen, Kottakkaran Sooppy Nisar. On generalized k-fractional derivative operator[J]. AIMS Mathematics, 2020, 5(3): 1936-1945. doi: 10.3934/math.2020129

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  • The principal aim of this paper is to introduce k-fractional derivative operator by using the definition of k-beta function. This paper establishes some results related to the newly defined fractional operator such as the Mellin transform and the relations to k-hypergeometric and k-Appell's functions. Also, we investigate the k-fractional derivative of k-Mittag-Leffler and the Wright hypergeometric functions.


    The classical beta function

    B(δ1,δ2)=0tδ11(1t)δ21dt,((δ1)>0,(δ2)>0) (1.1)

    and its relation with well known gamma function is given by

    B(δ1,δ2)=Γ(δ1)Γ(δ2)Γ(δ1+δ2),(δ1)>0,(δ2)>0.

    The Gauss hypergeometric, confluent hypergeometric and Appell's functions which are respectively defined by(see [27])

    2F1(δ1,δ2;δ3;z)=n=0(δ1)n(δ2)n(δ3)nznn!,(|z|<1),    (δ1,δ2,δ3C  and  δ30,1,2,3,), (1.2)

    and

    1Φ1(δ2;δ3;z)=n=0(δ2)n(δ3)nznn!,(|z|<1),    (δ2,δ3C  and  δ30,1,2,3,). (1.3)

    The Appell's series or bivariate hypergeometric series is defined by

    F1(δ1,δ2,δ3;δ4;x,y)=m,n=0(δ1)m+n(δ2)m(δ3)nxmyn(δ4)m+nm!n!; (1.4)

    for all δ1,δ2,δ3,δ4C,δ40,1,2,3,,|x|,|y|<1<1.

    The integral representation of hypergeometric, confluent hypergeometric and Appell's functions are respectively defined by

    2F1(δ1,δ2;δ3;z)=Γ(δ3)Γ(δ2)Γ(δ3δ2)10tδ21(1t)δ3δ21(1zt)δ1dt, (1.5)
    ((δ3)>(δ2)>0,|arg(1z)|<π),

    and

    1Φ1(δ2;δ3;z)=Γ(δ3)Γ(δ2)Γ(δ3δ2)10tδ21(1t)δ3δ21eztdt, (1.6)
    ((δ3)>(δ2)>0).
    F1(δ1,δ2,δ3;δ4;x,y)=Γ(δ4)Γ(δ1)Γ(δ4δ1)10tδ11(1t)δ4δ11(1xt)δ2(1yt)δ3dt. (1.7)

    The k-gamma function, k-beta function and the k-Pochhammer symbol introduced and studied by Diaz and Pariguan [5]. The integral representation of k-gamma function and k-beta function respectively given by

    Γk(z)=kzk1Γ(zk)=0tz1ezkkdt,(z)>0,k>0 (1.8)
    Bk(x,y)=1k10txk1(1t)yk1dt,(x)>0,(y)>0. (1.9)

    Here, we recall the following relations (see [5]).

    Bk(x,y)=Γk(x)Γk(y)Γk(x+y), (1.10)
    (z)n,k=Γk(z+nk)Γk(z), (1.11)

    where (z)n,k=(z)(z+k)(z+2k)(z+(n1)k);(z)0,k=1 and k>0

    and

    n=0(α)n,kznn!=(1kz)αk. (1.12)

    These studies were followed by Mansour [16], Kokologiannaki [13], Krasniqi [14] and Merovci [17]. In 2012, Mubeen and Habibullah [18] defined the k-hypergeometric function as

    2F1,k(δ1,δ2;δ3;z)=n=0(δ1)n,k(δ2)n,k(δ3)n,kznn!, (1.13)

    where δ1,δ2,δ3C and δ30,1,2, and its integral representation is given by

    2F1,k(δ1,δ2;δ3;z)=1kBk(δ2,δ3δ2)×10tδ2k1(1t)δ3δ2k1(1ktz)δ1kdt. (1.14)

    The k-Riemann-Liouville (R-L) fractional integral using k-gamma function introduced in [19]:

    (Iαkf(t))(x)=1kΓk(α)x0f(t)(xt)αk1dt,k,αR+. (1.15)

    Later on Mubeen and Iqbal [11] established the improved version of Gruss type inequalities by utilizing k-fractional integrals. In [1], Agarwal et al. presented certain Hermite-Hadamard type inequalities for generalized k-fractional integrals. Set et al. [29] presented an integral identity and generalized Hermite–Hadamard type inequalities for Riemann–Liouville fractional integral. Mubeen et al. [24] established integral inequalities of Ostrowski type for k-fractional Riemann–Liouville integrals. Recently, many researchers have introduced generalized version of k-fractional integrals and investigated a large bulk of various inequalities via the said fractional integrals. The interesting readers are referred to see the work of [9,10,26,30]. Farid et al. [7] introduced Hadamard k-fractional integrals. In [8] introduced Hadamard-type inequalities for k-fractional Riemann-Liouville integrals. In [12,31], the authors established certain inequalities by utilizing Hadamard-type inequalities for k-fractional Riemann-Liouville integrals. In [25], Nisar et al. established certain Gronwall type inequalities associated with Riemann-Liouville k- and Hadamard k-fractional derivatives and their applications. In [25], they presented dependence solutions of certain k-fractional differential equations of arbitrary real order with initial conditions. Recently, Samraiz et al. [28] defined an extension of Hadamard k-fractional derivative and proved its various properties.

    The solution of some integral equations involving confluent k-hypergeometric functions and k-analogue of Kummer's first formula are given in [22,23]. While the k-hypergeometric and confluent k-hypergeometric differential equations are introduced in [20]. In 2015, Mubeen et al. [21] introduced k-Appell hypergeometric function as

    F1,k(δ1,δ2,δ3;δ4;z1,z2)=m,n=0(δ1)m+n,k(δ2)m,k(δ3)m,k(δ4)m+n,kzm1zn2m!n! (1.16)

    for all δ1,δ2,δ3,δ4C,δ40,1,2,3,,max{|z1|,|z2|}<1k and k>0. Also, Mubeen et al. defined its integral representation as

    F1,k(δ1,δ2,δ3;δ4;z1,z2)=1kBk(δ1,δ4δ1)10tδ1k1(1t)δ4δ1k1(1kz1t)δ2k(1kz2t)δ3kdt, (1.17)
    ((δ4)>(δ1)>0).

    In this section, we recall the following definition of fractional derivatives from and give a new extension called Riemann-Liouville k-fractional derivative.

    Definition 2.1. The well-known R-L fractional derivative of order μ is defined by

    Dμx{f(x)}=1Γ(μ)x0f(t)(xt)μ1dt,(μ)<0. (2.1)

    For the case m1<(μ)<m where m=1,2,, it follows

    Dμx{f(x)}=dmdxmDμmx{f(x)}=dmdxm{1Γ(μ+m)x0f(t)(xt)μ+m1dt}. (2.2)

    For further study and applications, we refer the readers to the work of [2,3,4,15,32]. In the following, we define Riemann-Liouville k-fractional derivative of order μ as

    Definition 2.2.

    kDμx{f(x)}=1kΓk(μ)x0f(t)(xt)μk1dt,(μ)<0,kR+. (2.3)

    For the case m1<(μ)<m where m=1,2,, it follows

    kDμx{f(x)}=dmdxmkDμmkx{f(x)}=dmdxm{1kΓk(μ+mk)x0f(t)(xt)μk+m1dt}. (2.4)

    Note that for k=1, definition 2.2 reduces to the classical R-L fractional derivative operator given in definition 2.1.

    Now, we are ready to prove some theorems by using the new definition 2.2.

    Theorem 1. The following formula holds true,

    kDμz{zηk}=zημkΓk(μ)Bk(η+k,μ),(μ)<0. (2.5)

    Proof. From (2.3), we have

    kDμz{zηk}=1kΓk(μ)z0tηk(zt)μk1dt. (2.6)

    Substituting t=uz in (2.6), we get

    kDμz{zηk}=1kΓk(μ)10(uz)ηk(zuz)μk1zdu=zημkkΓk(μ)10uηk(1u)μk1du.

    Applying definition (1.9) to the above equation, we get the desired result.

    Theorem 2. Let (μ)>0 and suppose that the function f(z) is analytic at the origin with its Maclaurin expansion given by f(z)=n=0anzn where |z|<ρ for some ρR+. Then

    kDμz{f(z)}=n=0ankDμz{zn}. (2.7)

    Proof. Using the series expansion of the function f(z) in (2.3) gives

    kDμz{f(z)}=1kΓk(μ)z0n=0antn(zt)μk1dt.

    As the series is uniformly convergent on any closed disk centered at the origin with its radius smaller then ρ, therefore the series so does on the line segment from 0 to a fixed z for |z|<ρ. Thus it guarantee terms by terms integration as follows

    kDμz{f(z)}=n=0an{1kΓk(μ)z0tn(zt)μk1dt=n=0ankDμz{zn},

    which is the required proof.

    Theorem 3. The following result holds true:

    kDημz{zηk1(1kz)βk}=Γk(η)Γk(μ)zμk12F1,k(β,η;μ;z), (2.8)

    where (μ)>(η)>0 and |z|<1.

    Proof. By direct calculation, we have

    kDημz{zηk1(1kz)βk}=1kΓk(μη)z0tηk1(1kt)βk(zt)μηk1dt=zμηk1kΓk(μη)z0tηk1(1kt)βk(1tz)μηk1dt.

    Substituting t=zu in the above equation, we get

    kDημz{zηk1(1kz)βk}=zμk1kΓk(μη)10uηk1(1kuz)βk(1u)μηk1zdu.

    Applying (1.14) and after simplification we get the required proof.

    Theorem 4. The following result holds true:

    kDημz{zηk1(1kaz)αk(1kbz)βk}=Γk(η)Γk(μ)zμk1F1,k(η,α,β;μ;az,bz), (2.9)

    where (μ)>(η)>0, (α)>0, (β)>0, max{|az|,|bz|}<1k.

    Proof. To prove (2.9), we use the power series expansion

    (1kaz)αk(1kbz)βk=m=0n=0(α)m,k(β)n,k(az)mm!(bz)nn!.

    Now, applying Theorem 1, we obtain

    kDημz{zηk1(1kaz)αk(1kbz)βk}=m=0n=0(α)m,k(β)n,k(a)mm!(b)nn!kDημz{zηk+m+n1}=m=0n=0(α)m,k(β)n,k(a)mm!(b)nn!βk(η+mk+nk,μη)Γk(μη)zμk+m+n1=m=0n=0(α)m,k(β)n,k(a)mm!(b)nn!Γk(η+mk+nk)Γk(μ+mk+nk)zμk+m+n1.

    In view of (1.16), we get

    kDημz{zηk1(1kaz)αk(1kbz)βk}=Γk(η)Γk(μ)zμk1F1,k(η,α,β;μ;az,bz).

    Theorem 5. The following Mellin transform formula holds true:

    M{exkDμz(zηk);s}=Γ(s)Γk(μ)Bk(η+k,μ)zημk, (2.10)

    where (η)>1, (μ)<0, (s)>0.

    Proof. Applying the Mellin transform on definition (2.3), we have

    M{exkDμz(zηk);s}=0xs1exkDμz(zη);s}dx=1kΓk(μ)0xs1ex{z0tηk(zt)μk1dt}dx=zμk1kΓk(μ)0xs1ex{z0tηk(1tz)μk1dt}dx=zημkkΓk(μ)0xs1ex{10uηk(1u)μk1du}dx

    Interchanging the order of integrations in above equation, we get

    M{exkDμz(zηk);s}=zημkkΓk(μ)10uηk(1u)μk1(0xs1exdx)du.=zημkkΓk(μ)Γ(s)10uηk(1u)μk1du=Γ(s)Γk(μ)Bk(η+k,μ)zημk,

    which completes the proof.

    Theorem 6. The following Mellin transform formula holds true:

    M{exkDμz((1kz)αk);s}=zμkΓ(s)Γk(μ)Bk(k,μ)2F1,k(α,k;μ+k;z), (2.11)

    where (α)>0, (μ)<0, (s)>0, and |z|<1.

    Proof. Using the power series for (1kz)αk and applying Theorem 5 with η=nk, we can write

    M{exkDμz((1kz)αk);s}=n=0(α)n,kn!M{exkDμz(zn);s}=Γ(s)kΓk(μ)n=0(α)n,kn!Bk(nk+k,μ)znμk=Γ(s)zμkΓk(μ)n=0Bk(nk+k,μ)(α)n,kznn!=Γ(s)zμkn=0Γk(k+nk)Γk(μ+k+nk)(α)n,kznn!=Γ(s)Γk(μ+k)zμkn=0(k)n,k(μ+k)n,k(α)n,kznn!=Γ(s)zμkΓk(μ)Bk(k,μ)2F1,k(α,k;μ+k;z),

    which is the required proof.

    Theorem 7. The following result holds true:

    kDημz[zηk1Eμk,γ,δ(z)]=zμk1kΓk(μη)n=0(μ)n,kΓk(γn+δ)Bk(η+nk,μη)znn!, (2.12)

    where γ,δ,μC, (p)>0, (q)>0, (μ)>(η)>0 and Eμk,γ,δ(z) is k-Mittag-Leffler function (see [6]) defined as:

    Eμk,γ,δ(z)=n=0(μ)n,kΓk(γn+δ)znn!. (2.13)

    Proof. Using (2.13), the left-hand side of (2.12) can be written as

    kDημz[zηk1Eμk,γ,δ(z)]=kDημz[zηk1{n=0(μ)n,kΓk(γn+δ)znn!}].

    By Theorem 2, we have

    kDημz[zηk1Eμk,γ,δ(z)]=n=0(μ)n,kΓk(γn+δ){kDμz[zηk+n1]}.

    In view of Theorem 1, we get the required proof.

    Theorem 8. The following result holds true:

    kDημz{zηk1mΨn[(αi,Ai)1,m;|z(βj,Bj)1,n;]}=zμk1kΓk(μη)×n=0mi=1Γ(αi+Ain)nj=1Γ(βj+BjnBk(η+nk,μη)znn!, (2.14)

    where (p)>0, (q)>0, (μ)>(η)>0 and mΨn(z) is the Fox-Wright function defined by (see [15], pages 56–58)

    mΨn(z)=mΨn[(αi,Ai)1,m;|z(βj,Bj)1,n;]=n=0mi=1Γ(αi+Ain)nj=1Γ(βj+Bjnznn!. (2.15)

    Proof. Applying Theorem 1 and followed the same procedure used in Theorem 7, we get the desired result.

    Recently, many researchers have introduced various generalizations of fractional integrals and derivatives. In this line, we have established a k-fractional derivative and its various properties. If we letting k1 then all the results established in this paper will reduce to the results related to the classical Reimann-Liouville fractional derivative operator.

    The author K.S. Nisar thanks to Deanship of Scientific Research (DSR), Prince Sattam bin Abdulaziz University for providing facilities and support.

    The authors declare no conflict of interest.



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