Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


IMPACT FACTOR 2018: 0.726
5-year IMPACT FACTOR: 0.869

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2018: 0.34

ICV 2017: 161.82

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 15, Issue 1

Issues

Volume 13 (2015)

Extended Riemann-Liouville type fractional derivative operator with applications

P. Agarwal / Juan J. Nieto
  • Corresponding author
  • Departamento de Estatística, Análise Matemática e Optimización, Instituto de Matemáticasd, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ M.-J. Luo
Published Online: 2017-12-29 | DOI: https://doi.org/10.1515/math-2017-0137

Abstract

The main purpose of this paper is to introduce a class of new extended forms of the beta function, Gauss hypergeometric function and Appell-Lauricella hypergeometric functions by means of the modified Bessel function of the third kind. Some typical generating relations for these extended hypergeometric functions are obtained by defining the extension of the Riemann-Liouville fractional derivative operator. Their connections with elementary functions and Fox’s H-function are also presented.

Keywords: Gamma function; Extended beta function; Riemann-Liouville fractional derivative; Hypergeometric functions; Fox H-function; Generating functions; Mellin transform; Integral representations

MSC 2010: 26A33; 33B15; 33B20; 33C05; 33C20; 33C65

1 Introduction

Extensions and generalizations of some known special functions are important both from the theoretical and applied point of view. Also many extensions of fractional derivative operators have been developed and applied by many authors (see [2, 3, 4, 5, 6, 11, 12, 19, 20, 21] and [17, 18]). These new extensions have proved to be very useful in various fields such as physics, engineering, statistics, actuarial sciences, economics, finance, survival analysis, life testing and telecommunications. The above-mentioned applications have largely motivated our present study.

The extended incomplete gamma functions constructed by using the exponential function are defined by α,z;p=0ztα1exptptdt(p)>0;p=0,(α)>0(1)

and Γα,z;p=ztα1exptptdt(p)0(2)

with |arg z| < π, which have been studied in detail by Chaudhry and Zubair (see, for example, [2] and [4]). The extended incomplete gamma functions γ(α, z; p) and Γ(α, z; p) satisfy the following decomposition formula α,z;p+Γα,z;p=Γpα=2pα/2Kα(2p)(p)>0,(3)

where Γp(α) is called extended gamma function, and Kα(z) is the modified Bessel function of the third kind, or the Macdonald function with its integral representation given by (see [7]) Kα(z)=120expκ(z|t)dttα+1,(4)

where ℜ(z) > 0 and κ(z|t)=z2t+1t.(5)

For α = 12, we have K12(z)=π2zez.(6)

Instead of using the exponential function, Chaudhry and Zubair extended (1) and (2) in the following form (see [3], see also [4]) γμα,x;p=2pπ0xtα32exptKμ+12ptdt(7)

and Γμα,x;p=2pπxtα32exptKμ+12ptdt,(8)

where ℜ(x) > 0, ℜ(p) > 0, −∞ < α < ∞.

Inspired by their construction of (7) and (8), we aim to introduce a class of new special functions and fractional derivative operator by suitably using the modified Bessel function Kα(z).

The present paper is organized as follows: In Section 2, we first define the extended beta function and study some of its properties such as different integral representations and its Mellin transform. Then some extended hypergeometric functions are introduced by using the extended beta function. The extended Riemann-Liouville type fractional derivative operator and its properties are given in Section 3. In Section 4, the linear and bilinear generating relations for the extended hypergeometric functions are derived. Finally, the Mellin transforms of the extended fractional derivative operator are determined in Section 5.

2 Extended beta and hypergeometric functions

This section is divided into two subsections. In subsection-1, we define the extended beta function Bμ(x, y; p; m) and study some of its properties. In subsection-2, we introduce the extended Gauss hypergeometric function Fμ(a, b; c; z; p; m), the Appell hypergeometric functions F1,μ, F2,μ and the Lauricella hypergeometric function F3,μD and then obtain their integral representations. Throughout the present study, we shall assume that ℜ(p) > 0 and m > 0.

2.1 Extended beta function

Definition 2.1

The extended beta function Bμ(x, y; p; m) with ℜ(p) > 0 is defined by Bμ(x,y;p;m):=2pπ01tx32(1t)y32Kμ+12ptm(1t)mdt,(9)

where x, y ∈ ℂ, m > 0 and ℜ(μ) ≥ 0.

Remark 2.2

Taking m = 1, μ = 0 and making use of (6), (9) reduces to the extended beta function Bμ(x, y; p) defined by Chaudhry et al. [5, Eq. (1.7)] B(x,y;p):=B0(x,y;p;1)=01tx11ty1exppt(1t)dt,(10)

where ℜ(p) > 0 and x, y ∈ ℂ.

Theorem 2.3

The following integral representations for the extended beta functions Bμ(x, y; p; m) with ℜ(p) > 0 are valid Bμ(x,y;p;m)=22pπ0π2cos2(x1)θsin2(y1)θKμ+12psec2mθcsc2mθdθ(11) =2pπ0ux32(1+u)x+y1Kμ+12p(1+u)2mumdu(12) =22xy2pπ11(1+u)x32(1u)y32Kμ+1222mp(1u2)mdu.(13)

Proof

These formulas can be obtained by using the transformations t=cos2θ,t=u1+uand t=1+u2 in (9), respectively.□

Theorem 2.4

The following expression holds true Bμx,y;p,m=122pπ0B0xm2,ym2;κp|u;mduuμ+32,x,yC,m>0,p>0,μ0(14)

where κ(p|u) is given by (5).

Proof

Expressing Bμ(x, y; p, m) in its integral form with the help of (9), and taking (4) into account, we obtain Bμx,y;p,m=2pπ01tx321ty32Kμ+12ptm1tmdt=122pπ01tx321ty320expκp|utm1tmduuμ+32dt=122pπ001tx321ty32expκp|utm1tmdtduuμ+32,(15)

where κ(p|u) is given by (5).

In order to write the inner integral as our extended beta function, we need the following variant of (6), that is, ez=2πzK12z.

Then, we have 01tx321ty32expκp|utm1tmdt=01tx321ty322πκp|utm1tmK12κp|utm1tmdt=2κp|uπ01txm2321tym232K12κp|utm1tmdt=B0xm2,ym2;κp|u;m.(16)

Substituting (16) into (15) we obtain the required result (14).□

Remark 2.5

It is interesting to note that using the definition of the extended beta function (see, [11, 12]) we can get the following expression for (9) Bμx,y;p,m=122pπ0Bκp|u;m,mx12,y12duuμ+32,(17)

where the function Bb;ρ,λ(x, y) is given by [12, p. 631, Eq. (2)] Bb;ρ,λx,y=01tx11ty1expbtρ1tλdtx,yC.

The following theorem establishes the relation between the Mellin transform Mf(x);xs=0xs1fxdx

and the extended beta function.

Theorem 2.6

Let x, y ∈ ℂ, m > 0, ℜ(μ) ≥ 0 and (s)>max(μ),12+12mxm,12+12mym.

Then we have the following relation MBμx,y;p;m:ps=12μΓs+μΓs2μ2Γx+ms+m12Γy+ms+m12Γs2+μ2Γx+y+2ms+m1=12μΓs+μΓs2μ2Γs2+μ2Bx+ms+m12,y+ms+m12.(18)

Proof

First, we have MBμx,y;p;m:ps=0ps12pπ01tx321ty32Kμ+12ptm(1t)mdtdp=2π01tx321ty320ps+121Kμ+12ptm(1t)mdpdt=2π01tx+ms+12321ty+ms+1232dt0us+121Kμ+12udu=2πΓx+ms+m12Γy+ms+m12Γx+y+2ms+m10us+121Kμ+12udu.(19)

Since the Mellin transform of the Macdonald function Kv(z) is given by [9, p. 37, Eq.(1.7.41)]: MKvz:zs=2s2Γs2+v2Γs2v2,(20)

the last integral in (19) can be evaluated as 0us+121Kμ+12udu=2s32Γs2+μ2+12Γs2μ2=212μπΓs+μΓs2μ2Γs2+μ2,(21)

where we have used Γ2z=22z1πΓzΓz+12.

Finally, we get MBμx,y;p;m:ps=12μΓs+μΓs2μ2Γx+ms+m12Γy+ms+m12Γs2+μ2Γx+y+2ms+m1.

Now we can derive the Fox H-function representation of the extended beta function defined in (9).

Let m, n, p, q be integers such that 0 ≥ mq, 0 ≥ np, and for parameters ai, bi ∈ ℂ and for parameters αi, βj ∈ ℝ+ (i = 1, …, p; j = 1, …, q), the H-function is defined in terms of a Mellin-Barnes integral in the following manner ([8, pp. 1–2]; see also [10, p. 343, Definition E.1.] and [13, p. 2, Definition 1.1.]): Hp,qm,nzai,αi1,p(bj,βj)1,q=Hp,qm,nza1,α1,,ap,αp(b1,β1),,(bq,βq)=12πiLΘszsds,(22)

where Θ(s)=j=1mΓ(bj+βjs)i=1nΓ(1aiαis)i=n+1pΓ(ai+αis)j=m+1qΓ(1bjβjs),(23)

with the contour 𝓛 suitably chosen. As convention, the empty product is equal to one. The theory of the H-function is well explained in the books of Mathai [14], Mathai and Saxena ([15], Ch.2), Srivastava, Gupta and Goyal ([22], Ch.1) and Kilbas and Saigo ([8], Ch.1 and Ch.2). Note that (18) and (20) mean Hp,qm,n (z) in (22) is the inverse Mellin transform of Θ(s) in (23).

Theorem 2.7

Let ℜ(p) > 0, x, y ∈ ℂ, m > 0 and ℜ(μ) ≥ 0, then Bμx,y;p;m=12μH2,44,0p(μ2,12),(x+y+m1,2m)μ,1,(μ2,12),(x+m12,m),(y+m12,m),

where Bμ(x, y; p; m) is as defined in (9).

Proof

The result is obtained by taking the inverse Mellin transform of (18) in Theorem 2.6 and using (22) and (23).□

2.2 Extended hypergeometric functions

Definition 2.8

The extended Gauss hypergeometric function Fμ(a, b; c; z; p; m) is defined by Fμ(a,b;c;z;p;m):=n=0(a)nBμ(b+n,cb;p;m)B(b,cb)znn!,(24)

where ℜ(p) > 0, ℜ(μ) ≥ 0, 0 < ℜ(b) < ℜ(c) and |z| < 1.

Definition 2.9

The extended Appell hypergeometric function F1,μ is defined by F1,μ(a,b,c;d;x,y;p;m):=n,k=0(b)n(c)kBμ(a+n+k,da;p;m)B(a,da)xnn!ykk!,(25)

where ℜ(p) > 0, ℜ(μ) ≥ 0, 0 < ℜ(a) < ℜ(d) and |x| < 1, |y| < 1.

Definition 2.10

The extended Appell hypergeometric function F2,μ is defined by F2,μa,b,c;d,e;x,y;p;m:=n,k=0(a)n+kBμ(b+n,db;p;m)B(b,db)Bμ(c+k,ec;p;m)B(c,ec)xnn!ykk!,(26)

where ℜ(p) > 0, ℜ(μ) ≥ 0, 0 < ℜ(b) < ℜ(d), 0 < ℜ(c) < ℜ(e) and |x| + |y| < 1.

Definition 2.11

The extended Lauricella hypergeometric function FD,μ3 is FD,μ3(a,b,c,d;e;x,y,z;p;m):=n,k,r=0(b)n(c)k(d)rBμ(a+n+k+r,ea;p;m)B(a,ea)xnn!ykk!zrr!,(27)

where ℜ(p) > 0, ℜ(μ) ≥ 0, 0 < ℜ(a) < ℜ(e) and |x| < 1, |y| < 1, |z| < 1.

Here, it is important to mention that when we take m = 1, μ = 0 and then letting p → 0, function (24) reduces to the ordinary Gauss hypergeometric function defined by 2F1a,b;c;z2F1a,bc;z=n=0an(b)ncnznn!,(28)

where (x)n denotes the Pochhammer symbol defined, in terms of the familiar gamma function, by (x)n=Γ(x+n)Γ(x)=1n=0;xC{0}xx+1x+n1nN;xC.

For conditions of convergence and other related details of this function, see [1], [9] and [16]. Similarly, we can reduce the functions (25), (26) and (27) to the well-known Appell functions F1, F2 and Lauricella function FD3, respectively (see [16] and [23]).

Now, we establish the integral representations of the extended hypergeometric functions given by (24), (25), (26) and (27) as follows.

Theorem 2.12

The following integral representation for the extended Gauss hypergeometric function Fμ(a, b; c; z; p; m) is valid Fμa,b;c;z;p;m=2pπ1B(b,cb)01tb321tcb321ztaKμ+12ptm(1t)mdt,(29)

where |arg (1 − z)| < π, ℜ(p) > 0, m > 0 and ℜ(μ) ≥ 0.

Proof

By using (9) and employing the binomial expansion (1zt)a=n=0(a)n(zt)nn!|zt|<1,(30)

we get the above integral representation.□

Theorem 2.13

The following integral representation for the extended hypergeometric function F1,μ is valid F1,μ(a,b,c;d;x,y;p;m)=2pπ1B(a,da)×01ta321tda321xtb1ytcKμ+12ptm(1t)mdt,(31)

Proof

For simplicity, let ℑ denote the left-hand side of (31). Then, using (25) yields I=n,k=0(b)n(c)kBμ(a+n+k,da;p;m)B(a,da)xnn!ykk!,(32)

By applying (9) to the integrand of (31), after a little simplification, we have I=n,k=02pπ01ta+n+k32(1t)da32Kμ+12ptm(1t)mdt(b)n(c)kB(a,da)xnn!ykk!.(33)

By interchanging the order of summation and integration in (33), we get I=2pπ1B(a,da)01ta32(1t)da32Kμ+12ptm(1t)m×n=0(b)nn!(xt)nk=0(c)kk!(yt)kdt=2pπ1B(a,da)01ta32(1t)da32×1xtb1ytcKμ+12ptm(1t)mdt,(34)

which proves the integral representation (31).□

To establish Theorem 2.13, we need to recall the following elementary series identity involving the bounded sequence of {f(N)}N=0 stated in the following result.

Lemma 2.14

For a bounded sequence {f(N)}N=0 of essentially arbitrary complex numbers, we have N=0f(N)(x+y)NN!=n=0k=0f(n+k)xnn!ykk!.(35)

Theorem 2.15

The following integral representation for the extended hypergeometric function F2,μ is valid F2,μ(a,b,c;d,e;x,y;p;m)=2pπ1Bb,dbBc,ec×0101tb321tdb32wb321wec321xtywa×Kμ+12ptm(1t)mKμ+12pwm(1w)mdtdw.(36)

Proof

Let 𝓛 denote the left-hand side of (36). Then, using (26) yields L=n,k=0(a)n+kBμ(b+n,db;p;m)B(b,db)Bμ(c+k,ec;p;m)B(c,ec)xnn!ykk!.(37)

By applying (9) to the integrand of (32), we have L=2pπn,k=001tb+n321tdb32Kμ+12ptm(1t)mdt×01wb+n32(1w)ec32Kμ+12pwm(1w)mdw×(a)n+kB(b,db)B(c,ec)xnn!ykk!.(38)

Next, interchanging the order of summation and integration in (38), which is guaranteed, yields L=2pπ1Bb,dbBc,ec0101tb321tdb32wb321wec32×Kμ+12ptm(1t)mKμ+12pwm(1w)m×n,k=0(a)n+k(xt)nn!(yw)kk!dtdw.(39)

Finally, applying (35) to the double series in (39), we obtain the right-hand side of (36).□

Theorem 2.16

The following integral representation for the extended hypergeometric function FD,μ3 is valid FD,μ3(a,b,c,d;e;x,y,z;p;m)=1B(a,ea)2pπ×01ta32(1t)ea32(1xt)b1ytc1ztdKμ+12ptm(1t)mdt(40)

Proof

A similar argument in the proof of Theorem 2.15 will be able to establish the integral representation in (40). Therefore, details of the proof are omitted.□

3 Extended Riemann-Liouville fractional derivative operator

We first recall that the classical Riemann-Liouville fractional derivative is defined by (see [23, p. 286]) Dzνf(z):=1Γ(ν)0z(zt)ν1f(t)dt,

where ℜ(ν) < 0 and the integration path is a line from 0 to z in the complex t-plane. It coincides with the fractional integral of order −ν. In case m − 1 < ℜ(ν) < m, m ∈ ℕ, it is customary to write Dzνf(z):=dmdzmDzνmf(z)=dmdzm1Γ(mν)0z(zt)mν1f(t)dt.

We present the following new extended Riemann-Liouville-type fractional derivative operator.

Definition 3.1

The extended Riemann-Liouville fractional derivative is defined as Dzν,μ;p;mf(z):=1Γ(ν)2pπ0z(zt)ν1f(t)Kμ+12pz2mtm(zt)mdt,(41)

where ℜ(ν) < 0, ℜ(p) > 0, ℜ(m) > 0 and ℜ(μ) ≥ 0.

For n − 1 < ℜ(ν) < n, n ∈ ℕ, we write Dzν,μ;p;mf(z):=dndznDzνn,μ;p;mf(z)=dndzn1Γ(nν)2pπ0z(zt)nν1f(t)Kμ+12pz2mtm(zt)mdt.(42)

Remark 3.2

If we take m = 0, μ = 0, and p → 0, then the above extended Riemann-Liouville fractional derivative operator reduces to the classical Riemann-Liouville fractional derivative operator.

Now, we begin our investigation by calculating the extended fractional derivatives of some elementary functions. For our purpose, we first establish two results involving the extended Riemann-Liouville fractional derivative operator.

Lemma 3.3

Let ℜ(ν) < 0, then we have Dzν,μ;p;mzλ=zλνΓ(ν)Bμλ+32,ν+12;p;m.(43)

Proof

Using Definition 3.1 and 1, we have Dzν,μ;p;mzλ=1Γ(ν)2pπ0z(zt)ν1tλKμ+12pz2mtm(zt)mdt=zλνΓ(ν)2pπ01(1u)ν+1232uλ+3232Kμ+12pum(1u)mdu=zλνΓ(ν)Bμλ+32,ν+12;p;m.

Next, we apply the extended Riemann-Liouville fractional derivative to a function f(z) analytic at the origin.

Lemma 3.4

Let ℜ(ν) < 0 and suppose that a function f(z) is analytic at the origin with its Maclaurin expansion given by f(z)=n=0anzn (|z| < ρ) for some ρ ∈ ℝ+. Then we have Dzν,μ;p;mf(z)=n=0anDzν,μ;p;mzn.

Proof

Using Definition 3.1 to the function f(z) with its series expansion, we have Dzν,μ;p;mf(z)=1Γ(ν)2pπ0z(zt)ν1Kμ+12pz2mtm(zt)mn=0antndt.

Since the power series converges uniformly on any closed disk centered at the origin with its radius smaller than ρ, so does the series on the line segment from 0 to a fixed z for |z| < ρ. This fact guarantees term-by-term integration as follows: Dzν,μ;p;mf(z)=n=0an{1Γ(ν)2pπ0z(zt)ν1Kμ+12pz2mtm(zt)mtndt}=n=0anDzν,μ;p;mzn.

As a consequence we have the following result.

Theorem 3.5

Let ℜ(ν) < 0 and suppose that a function f(z) is analytic at the origin with its Maclaurin expansion given by f(z)=n=0anzn (|z| < ρ) for some ρ ∈ ℝ+. Then we have Dzν,μ;p;mzλ1f(z)=n=0anDzν,μ;p;mzλ+n1=zλν1Γ(ν)n=0anBμλ+n+12,ν+12;p;mzn.

We present two subsequent theorems which may be useful to find certain generating function.

Theorem 3.6

For ℜ(ν) > ℜ(λ) > − 12, we have Dzλν,μ;p;m{zλ1(1z)α}=zν1Γ(νλ)Bλ+12,νλ+12Fμα,λ+12;ν+1;z;p;m(|z|<1;αC).(44)

Proof

Using (30) and applying Lemmas 3.3 and 3.4, we obtain Dzλν,μ;p;m{zλ1(1z)α}=Dzλν,μ;p;mzλ1l=0(α)lzll!=l=0(α)ll!Dzλν,μ;p;mzλ+l1=l=0(α)ll!Bμ(λ+l+12,νλ+12;p;m)Γ(νλ)zν+l1.

By using (24), we can get Dzλν,μ;p;m{zλ1(1z)α}=zν1Γ(νλ)Bλ+12,νλ+12Fμα,λ+12;ν+1;z;p;m.

Theorem 3.7

Let ℜ(ν) > ℜ(λ) > − 12, ℜ(α) > 0, ℜ(β) > 0; |az| < 1 and |bz| < 1. Then we have Dzλν,μ;p;m{zλ1(1az)α(1bz)β}=zν1Γ(νλ)Bλ+12,νλ+12F1,μλ+12,α,β;ν+1;az,bz;p;m.(45)

Proof

Use the binomial theorem for (1 − az)α and (1 − bz)β. Apply Lemmas 3.3 and 3.4 to obtain Dzλν,μ;p;m{zλ1(1az)α(1bz)β}=Dzλν,μ;p;mzλ1l=0k=0(α)l(β)k(az)ll!(bz)kk!=l,k=0(α)l(β)kDzλν,μ;p;mzλ+l+k1(a)ll!(b)kk!=zν1l,k=0(α)l(β)kBμλ+l+k+12,νλ+12;p;mΓ(νλ)(az)nl!(bz)kk!.

By using (25), we get Dzλν,μ;p;m{zλ1(1az)α(1bz)β}=zν1Γ(νλ)Bλ+12,νλ+12F1,μλ+12,α,β;ν+1;az,bz;p;m.

Theorem 3.8

Let ℜ(ν) > ℜ(λ) > −12, ℜ(α) > 0, ℜ(β) > 0, ℜ(γ) > 0, |az| < 1, |bz| < 1 and |cz| < 1. Then we have Dzλν,μ;p;m{zλ1(1az)α(1bz)β(1cz)γ}=zν1Γ(νλ)×Bλ+12,νλ+12FD,μ3λ+12,α,β,γ;ν+1;az,bz,cz;p;m.(46)

Proof

As in the proof of Theorem 3.7, taking the binomial theorem for (1 − az)α, (1 − bz)β and (1 − cz)γ and applying Lemmas 3.3 and 3.4 and taking Definition 5 into account, one can easily prove Theorem 3.8.□

Theorem 3.9

Let ℜ(ν) > ℜ(λ) > − 12, ℜ(α) > 0, ℜ(γ) > ℜ(β) > 0; |x1z| < 1 and |x| + |z| < 1. Then we have Dzλν,μ;p;m{zλ1(1z)αFμα,β;γ;x1z;p;m}=zν1Bλ+12,νλ+12ΓνλF2,μα,β,λ+12;γ,ν+1;x,z;p;m(47)

Proof

By using (30) and applying the Definition 2.8 for Fμ, we get Dzλν,μ;p;mzλ1(1z)αFμα,β;γ;x1z;p;m=Dzλν,μ;p;mzλ1(1z)αn=0(α)nn!Bμ(β+n,γβ;p;m)B(β,γβ)x1zn=n=0(α)nBμ(β+n,γβ;p;m)B(β,γβ)Dzλν,μ;p;mzλ1(1z)αnxnn!.

Using Theorem 3.6 for Dzλν,μ;p;m{zλ1(1z)αn} and interpreting the extended hypergeometric function Fμ as its series representation, we get Dzλν,μ;p;mzλ1(1z)αFμα,β;γ;x1z;p;m=zν1Γ(νλ)Bλ+12,νλ+12n,k=0{(α)n+kBμβ+n,γβ;p;mBβ,γβ×Bμλ+k+12,νλ+12;p;mBλ+12,νλ+12xnn!zkk!}=zν1Γ(νλ)Bλ+12,νλ+12F2,μα,β,λ+12;γ,ν+1;x,z;p;m.

This completes the proof.□

4 Generating functions involving the extended Gauss hypergeometric function

Here, we establish some linear and bilinear generating relations for the extended hypergeometric function Fμ by using Theorems 3.6, 3.7 and 3.9.

Theorem 4.1

Let ℜ(λ) > 0 and ℜ(β) > ℜ(α) > −12. Then we have n=0(λ)nn!Fμλ+n,α+12;β+1;z;p;mtn=1tλFμλ,α+12;β+1;z1t;p;m.|z|<min{1,|1t|}(48)

Proof

We start by recalling the elementary identity [(1z)t]λ=(1t)λ1z1tλ

and expand its left-hand side to obtain (1z)λn=0(λ)nn!t1zn=(1t)λ1z1tλ(|t|<|1z|).

Multiplying both sides of the above equality by zα−1 and applying the extended Riemann-Liouville fractional derivative operator Dzαβ,μ;p;m on both sides, we find Dzαβ;μ;p;mn=0(λ)ntnn!zα1(1z)λn=Dzαβ;μ;p;m(1t)λzα11z1tλ.

Uniform convergence of the involved series makes it possible to exchange the summation and the fractional operator to give n=0(λ)nn!Dzαβ;μ;p;mzα1(1z)λntn=1tλDzαβ;μ;p;mzα11z1tλ.

The result then follows by applying Theorem 3.6 to both sides of the last identity.□

Theorem 4.2

Let ℜ(λ) > 0, ℜ(γ) > 0 and ℜ(β) > ℜ(α) > −12. Then we have n=0(λ)nn!Fμγn,α+12;β+1;z;p;mtn=1tλF1,μα+12,γ,λ;β+1;z;zt1t;p;m.|z|<1;|t|<|1z|;|z||t|<|1t|

Proof

Considering the following identity [1(1z)t]λ=(1t)λ1+zt1tλ

and expanding its left-hand side as a power series, we get n=0(λ)nn!(1z)ntn=(1t)λ1zt1tλ(|t|<|1z|).

Multiplying both sides by zα−1(1 − z)γ and applying the definition of the extended Riemann-Liouville fractional derivative operator Dzαβ,μ;p;m on both sides, we find Dzαβ;μ;p;mn=0(λ)nn!zα1(1z)γ(1z)ntn=Dzαβ;μ;p;m(1t)λzα1(1z)γ1zt1tλ.

The given conditions are found to allow us to exchange the order of the summation and the fractional derivative to yield n=0(λ)nn!Dzαβ;μ;p;mzα1(1z)γ+ntn=(1t)λDzαβ;μ;p;mzα1(1z)γ1zt1tλ.

Finally the result follows by using Theorems 3.6 and 3.7.□

Theorem 4.3

Let ξ>δ>12,β>α>12 and ℜ(λ) > 0. Then we have n=0(λ)nn!Fμλ+n,α+12;β+1;z;p;mFμn,δ+12;ξ+1;u;p;mtn=1tλF2,μλ,α+12,δ+12;β+1,ξ+1;z1t,ut1t;p;m.|z|<1;1u1zt<1;z1t+ut1t<1

Proof

Replacing t by (1 − u)t in (48) and multiplying both sides of the resulting identity by uδ−1 gives n=0(λ)nn!Fμλ+n,α+12;β+1;z;p;muδ11untn=uδ111utλFμλ,α+12;β+1;z1(1u)t;p;m.λ>0,β>α>12

Applying the fractional derivative Duδξ,μ;p;m to both sides of the resulting identity and changing the order of the summation and the fractional derivative yields n=0(λ)nn!Fμλ+n,α+12;β+1;z;p;mDuδξ,μ;p;muδ1(1u)ntn=Duδξ,μ;p;muδ1[1(1u)t]λFμλ,α+12;β+1;z1(1u)t;p;m.|(1u)t|<1;|ut|<|1t|

The last identity can be written as follows: n=0(λ)nn!Fμλ+n,α+12;β+1;z;p;mDuδξ,μ;p;muδ1(1u)ntn=1tλDuδξ,μ;p;muδ11ut1tλFμλ,α+12;β+1;z1t1ut1t;p;m.

Finally the use of Theorems 3.6 and 3.9 in the resulting identity is seen to give the desired result.□

5 Mellin transforms and further results

In this section, we first obtain the Mellin transform of the extended Riemann-Liouville fractional derivative operator.

Theorem 5.1

Let ℜ(ν) < 0, m > 0, ℜ(μ) ≥ 0 and s>maxμ,121mλm,12+νm.

Then we have the following relation MDzν,μ;p;mzλ:s=zλν2μΓ(ν)Γ(s+μ)Γs2μ2Γλ+ms+m2+1Γν+ms+m2Γs2+μ2Γλν+2ms+m+1=zλν2μΓ(ν)Γs+μΓs2μ2Γs2+μ2Bλ+ms+m2+1,ν+ms+m2.(49)

Proof

Taking the Mellin transform and using Lemma 3.3, we have M[Dzν,μ;p;mzλ:s]=0ps1Dzν,μ;p;mzλdp=zλνΓ(ν)0ps1Bμλ+32,ν+12;p;mdp.

Applying Theorem 2.6 to the last integral yields the desired result.□

Theorem 5.2

Let ℜ(ν) < 0, m > 0, ℜ(μ) ≥ 0, |z| < 1 and (s)>maxμ,121m,12+νm.

Then we have the following relation MDzν,μ;p;m1zα:s=zν2μΓ(ν)Γs+μΓs2μ2Γs2+μ2Bms+m2+1,ν+ms+m2×2F1α,ms+m2+1;ν+2ms+m+1;z,(50)

where 2F1 is a well known Gauss hypergeometric function given by (28).

Proof

Using the binomial series for (1 − z)λ and Theorem 5.1 with λ = n yields M[Dzν,μ;p;m{(1z)α}:s]=MDzν,μ;p;mn=0(α)nn!zn:s=n=0(α)nn!M[Dzν,μ;p;m{zn}:s]=n=0(α)nn!znν2μΓ(ν)Γs+μΓs2μ2Γn+ms+m2+1Γν+ms+m2Γs2+μ2Γnν+2ms+m+1.

Then the last expression is easily seen to be equal to the desired one.□

Now we present the extended Riemann-Liouville fractional derivative of zλ in terms of the Fox H-function.

Theorem 5.3

Let ℜ(p) > 0, ℜ(μ) ≥ 0, ℜ(ν) < 0 and m > 0. Then we have Dzν,μ;p;mzλ=zλν2μΓ(ν)H2,44,0p(μ2,12),(λν+m+1,2m)μ,1,(μ2,12),(λ+m2+1,m),(ν+m2,m).

Proof

The result can be obtained by taking the inverse Mellin transform of the result in Lemma 3.3 with the aid of Theorem 2.7.□

Applying the result in Theorem 3.3 to the Maclaurin series of ez and the series expression of the Gauss hypergeometric function 2F1 gives the extended Riemann-Liouville fractional derivatives of ez and 2F1 asserted by the following theorems.

Theorem 5.4

If ℜ(ν) < 0, then we have Dzν,μ;p;m{ez}=zνΓ(ν)n=0Bμn+32,ν+12;p;mznn!.

Theorem 5.5

If ℜ(ν) < 0, then we have Dzν,μ;p;m2F1(a,b;c;z)=zνΓ(ν)n=0(a)n(b)n(c)nBμn+32,ν+12;p;mznn!|z|<1.

Acknowledgement

The research of J.J. Nieto has been partially supported by the Ministerio de Economía y Competitividad of Spain under grants MTM2016–75140–P, MTM2013–43014–P, Xunta de Galicia, Grants GRC2015-004 and R2016-022, and co-financed by the European Community fund FEDER.

References

  • [1]

    G.E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge Univ. Press. 1999. Google Scholar

  • [2]

    M.A. Chaudhry and S.M. Zubair, Generalized incomplete gamma functions with applications, J. Comput. Appl. Math. 55 (1994), 99–124. CrossrefGoogle Scholar

  • [3]

    M.A. Chaudhry and S.M. Zubair, On an Extension of Generalized incomplete gamma functions with applications, J. Austral. Math. Soc. Ser. B 37(1996), 392–405. CrossrefGoogle Scholar

  • [4]

    M.A. Chaudhry and S.M. Zubair, On a Class of Incomplete Gamma Functions with Applications, Chapman and Hall (CRC Press Company), Boca Raton, London, New York and Washington, D.C., 2001. Google Scholar

  • [5]

    M.A. Chaudhry, A. Qadir, M. Rafique, and S.M. Zubair, Extension of Euler’s beta function, J. Comput. Appl. Math. 78 (1997) 19–32. CrossrefGoogle Scholar

  • [6]

    M.A. Chaudhry, A. Qadir, H.M. Srivastava, and R.B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput. 159 (2) (2004) 589–602. Google Scholar

  • [7]

    A. Erdéldyi, W. Magnus, F. Oberhettinger and F. G.Tricomi, Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953. Google Scholar

  • [8]

    A. A. Kilbas and M. Saigo, H-Transforms: Theory and Applications, Chapman and Hall (CRC Press Company), Boca Raton, London, New York and Washington, D.C., 2004. Google Scholar

  • [9]

    A.A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V, Amsterdam, 2006. Google Scholar

  • [10]

    V. Kiryakova, Generalized Fractional Calculus and Applications, Longman & J. Wiley, Harlow - N. York, 1994. Google Scholar

  • [11]

    D.M. Lee, A.K. Rathie, R.K. Parmar and Y.S. Kim, Generalization of extended beta function, hypergeometric and confluent hypergeometric functions, Honam Math. J. 33 (2) (2011) 187–206. CrossrefGoogle Scholar

  • [12]

    M. J. Luo, G. V. Milovanovic and P. Agarwal, Some results on the extended beta and extended hypergeometric functions, Appl. Math. Comput. 248(2014) 631–651. Web of ScienceGoogle Scholar

  • [13]

    A.M. Mathai, R.K. Saxena, and H.J. Haubold, The H-function: Theory and applications, Springer, New York, 2010. Google Scholar

  • [14]

    A.M. Mathai, A Handbook of Generalized Special Functions for Statistical and Physical Sciences, Clarendon Press, Oxford, 1993. Google Scholar

  • [15]

    A.M. Mathai and R.K. Saxena, The H-function with Applications in Statistics and Other Disciplines, Halsted Press [John Wiley & Sons], New York-London, Sydney, 1978. Google Scholar

  • [16]

    F.W.J. Olver, D.W. Lozier, R.F. Boisvert, and C.W. Clark (Eds.), NIST Handbook of Mathematical Functions, New York: Cambridge University Press. 2010. Google Scholar

  • [17]

    İ. O. Kiymaz, A. Četinkaya and P. Agarwal, An extension of Caputo fractional derivative operator and its applications, Journal of Nonlinear Sciene and Applications 9(6) (2016) 3611–3621 CrossrefGoogle Scholar

  • [18]

    İ. O. Kiymaz, P. Agarwal, S. Jain, A. Četinkaya, On a New Extension of Caputo Fractional Derivative Operator, Advances in Real and Complex Analysis with Applications, Birkhäuser, (2017) 261–275. Google Scholar

  • [19]

    M.A. Özarslan and E. Özergin, Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Math. Comput. Model. 52 (2010) 1825–1833. CrossrefWeb of ScienceGoogle Scholar

  • [20]

    E. Özergin, M.A. Özarslan, and A. Altin, Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math. 235 (2011) 4601–4610. Web of ScienceCrossrefGoogle Scholar

  • [21]

    M.A. Ragusa, A. Scapellato, Mixed Morrey spaces and its applications to partial differential equations, Nonlinear Analysis 151 (2017) 5165,  CrossrefGoogle Scholar

  • [22]

    H.M. Srivastava, K.C. Gupta and S.P. Goyal, The H-functions of One and Two Variables with Applications, South Asian Publishers, New Delhi, Madras, 1982. Google Scholar

  • [23]

    H.M. Srivastava and H.L. Manocha, A treatise on generating functions, Ellis Horwood Limited, Chichester, 1984. Google Scholar

About the article

Received: 2017-11-16

Accepted: 2017-11-30

Published Online: 2017-12-29


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 1667–1681, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0137.

Export Citation

© 2017 Agarwal et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Saïd Abbas, Mouffak Benchohra, Samira Hamani, and Johnny Henderson
Mathematica Moravica, 2019, Volume 23, Number 1, Page 107
[2]
Mehmet Ali Özarslan and Cemaliye Kürt
Advances in Difference Equations, 2019, Volume 2019, Number 1
[3]
Trang T Lê, Zach Osman, D K Watson, Martin Dunn, and B A McKinney
Physica Scripta, 2019, Volume 94, Number 6, Page 065203
[4]
Xia Liu, Tao Zhou, and Haiping Shi
Discrete Dynamics in Nature and Society, 2018, Volume 2018, Page 1
[5]
Silvério Rosa and Delfim F.M. Torres
Chaos, Solitons & Fractals, 2018, Volume 117, Page 142
[6]
Mohammad Hossein Heydari, Zakieh Avazzadeh, and Malih Farzi Haromi
Applied Mathematics and Computation, 2019, Volume 341, Page 215
[7]
Kamel Saoudi, Praveen Agarwal, Poom Kumam, Abdeljabbar Ghanmi, and Phatiphat Thounthong
Advances in Difference Equations, 2018, Volume 2018, Number 1
[8]
P. Agarwal, M. Chand, D. Baleanu, D. O’Regan, and Shilpi Jain
Advances in Difference Equations, 2018, Volume 2018, Number 1

Comments (0)

Please log in or register to comment.
Log in