Fractional Calculus of a k-Wrigth Type Function I Introduction ... - Hikari

Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 31, 1547 - 1557

Fractional Calculus of a k-Wrigth Type Function Luis Guillermo Romero and Ruben Alejandro Cerutti Faculty of Exact Sciences National University of Nordeste. Avda. Libertad 5540 (3400) Corrientes, Argentina [email protected] [email protected] Abstract γ This article deals with a function Wk,α,β (z) that in the case of γ = and k = 1 reduces to the classical Wright function Wα,β (z). γ (z) are presented and Some elementary properties of the new Wk,α,β it Laplace transform is obtained. Also it has been shown that the fractional Riemann-Liouville integral transform such functions with powers multipliers into functions of the same form with a very interesting relation between indices.

Mathematics Subject Classification: 26A33, 33E12 Keywords: k-Wrigth function, Laplace Transform, Fractional calculus

I

Introduction and Preliminaries

The importance of the role played by the Wright function in partial differential equation of fractional order is well known and was widely treated in papers by several authors including Gorenflo, Luchko, Mainardi (cf. [5]), Mainardi (cf. [7]), Mainardi, Pagnini (cf. [8]). Since Diaz and Pariguan (cf. [2]) introduced be the k-Pochhammer symbol and the k-Gamma function as a continuous deformation of the classic Euler Gamma function, many jobs issue dedicated to what you could appoint kcalculus have been published. Among them we highlight [2], [4], [9], [10]. In this article we introduce a generalization of the Wright function that we will named the k-Wright function, in whose definition is used the k-Gamma function Γk (z) and the k-Pochhammer symbol (γ)n,k and we will denoted by γ Wk,α,β (z). In Section 1 we present the elementary and essential definitions concerning to the fractional calculus which are necessary along the paper and introduce γ the series representation of the generalized k-Wright function Wk,α,β (z)

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L. G. Romero and R. A. Cerutti

In Section 2 we consider some elementary properties. Also we obtain it Laplace Transform whose expression contains the k-Mittag-Leffler function γ Ek,α,β (z). Finally in Section 3 we consider the auxiliary function E (t, k, α, β) and evaluate it Riemann-Liouville fractional integral and also it Riemann-Liouville derivative. In the development of this paper we use fractional integrals and fractional derivatives, and also Laplace transform, so we introduce the definitions and notations. Definition 1 Let f be a sufficiently well-behaved function with support in R+ , and let ν be a real number, ν > 0. The Riemann-Liouville fractional integral of order ν, I+ν f is given by I+ν f (t)

1 = Γ(ν)

t

0

(t − τ )ν−1 f (τ )dτ

(I.1)

where Γ(z) is the Euler Gamma function ∞ e−t tz−1 dt Γ(z) = 0

for Re(z) > 0, z a complex number. It is known that the semigroup property is verified I+ν I+η = I+η I+ν = I+ν+η

(I.2)

where by I+0 we denote the Identity operator (cf. [12]), (cf. [3]), (cf. [11]). ν is defined The Riemann-Liouville fractional derivative of order ν > 0, D+ as the left inverse of the Riemann-Liouville integral of order ν; i. e, ν ν D+ I+ = I, ν > 0 cf. [11]

(I.3)

Other way of defined this fractional derivative is the follow. Definition 2 Let ν be a real number, and let m be the integer sucht that m − 1 < ν ≤ m. Then the Riemann-Liouville fractional derivative of order ν is given by ν D+ f (t) = D m I+m−ν f (t)

(I.4)

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Fractional calculus of a k-Wrigth type function

Equivalently, we have ν f (t) = D+

dm 1 dtm Γ(m−ν) dm f (t) dtm

t

f (τ )dτ 0 (t−τ )ν+1−m

, m−1 0, Re(β) > 0, and (ξ)n,k denote the Pochhammer k-symbol defined by (cf. [1]) (ξ)n,k = ξ(ξ + k)(ξ + 2k)...(ξ + (n − 1)k); n = 1, 2, .... Easily it can be proved that when ξ = 1, and k = 1 (I.10) reduce to classical two parameters Mittag-Leffler function. It is known that the Wright function Wα,β is defined by the series

Wα,β =

∞ n=0

zn ; α > −1; β ∈ C n!Γ (αn + β)

(I.11)

where Γ(z) is the Euler Gamma function. By means of the series development a generalization of (I.12) is now introduced we put the following Definition 3 Let k ∈ R; α,β,γ∈ C, Re(α) > 0, Re(β) > 0. The k-Wrigth function is defined as γ Wk,α,β (z)

=

∞ n=0

zn (γ)n,k Γk (αn + β) (n!)2

(I.12)

where (γ)n,k is the k-Pochhammer symbol given by (γ)n,k = γ(γ +k)(γ +2k)...(γ +(n−1)k); n = 1, 2, ..., and Γk (z) is the k-Gamma function given by (I.9). γ Easily we can prove that Wk,α,β (z) → Wα,β (z) as k → 1 and γ = 1, because Γk (z) → Γ(z) and (γ)n,k → (γ)n

II II.1

Main Results Elementary Properties of the k-Wright function

In this section we obtain several elementary properties of our k-Wright γ (z) defined by (I.12) and some others associated with the kfunction Wk,α,β Mittag-Leffler function obtained by means of the Laplace Transform. Lemma 1 Let α,β,γ∈ C, Re(α) > 0, Re(β) > 0. Then there holds

γ γ Wk,α,β (z) = βWk,α,β+k (z) + αz

d γ W (z) dz k,α,β+k

(II.1)

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Proof. By definition (I.3) we have γ βWk,α,β+k (z) + αz

β

∞ n=0

d γ W (z) = dz k,α,β+k

∞ zn zn (γ)n,k (γ)n,k d + αz = Γk (αn + β + k) (n!)2 dz n=0 Γk (αn + β + k) (n!)2

∞ n=0

β(γ)n,k αz(γ)n,k zn nz n−1 + = Γk (αn + β + k) (n!)2 n=0 Γk (αn + β + k) (n!)2 ∞

∞ (αn + β) (γ)n,k z n Γ (αn + β + k) (n!)2 n=0 k

(II.2)

Taking into account Γk (z + k) = zΓk (z), (II.2) it result zn (αn + β) (γ)n,k d γ Wk,α,β+k (z) = = 2 dt (αn + β) Γ (αn + β) (n!) k n=0 ∞

γ (z) + αz βWk,α,β+k

∞ n=0

zn (γ)n,k γ = Wk,α,β (z) Γk (αn + β) (n!)2

Then, we have γ γ Wk,α,β (z) = βWk,α,β+k (z) + αz

d γ (z) W dz k,α,β+k

Lemma 2 Let α,β,γ∈ C, Re(α) > 0, Re(β) > 0. Then there holds the formula ∞ zn (γ + k)n,k d γ Wk,α,β (z) = γ dz Γ (αn + α + β) (n + 1)(n!)2 n=0 k

(II.3)

Proof. From definition (I.3) and using the well known relations for the kPochhammer symbol (γ)n+1,k = γ(γ + k)n,k we have ∞ zn d γ (γ)n,k d = Wk,α,β (z) = dz dz n=0 Γk (αn + β) (n!)2 ∞ n=1

nz n−1 (n + 1)z n (γ)n,k (γ)n+1,k = 2 = Γk (αn + β) (n!)2 Γ (α(n + 1) + β) [(n + 1)!] k n=0 ∞

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γ

∞

zn (γ + k)n,k Γk (αn + α + β) (n + 1)(n!)2

n=0

γ Furthermore, it can be proved that Wk,α,β (z) verified the following differential equation

k d γ γ+k γ (z) + Wk,α,β (z) Wk,α,β (z) = z Wk,α,β γ dz

Lemma 3 Let α,β,γ∈ C, Re(α) > 0, Re(β) > 0. Then

k d γ γ+k γ Wk,α,β (z) − Wk,α,β (z) = (z) z Wk,α,β γ dz

(II.4)

Proof. From (I.3) we have γ+k Wk,α,β (z)

−

γ Wk,α,β (z)

∞ (γ + k)n,k − (γ)n,k z n = Γk (αn + β) (n!)2 n=0

(II.5)

Taking into account that the Pochhammer k-symbol verified (γ + k)n,k = 1 + n γk (γ)n,k it result

k (γ + k)n,k − (γ)n,k = n (II.6) (γ)n,k γ Replacing (II.6) in (II.5) it result ∞

n

k γ

(γ)n,k z n = Γ (αn + β) (n!)2 n=0 k k ∞ ∞ n γ (γ)n,k z n (n + 1) γk (γ)n+1,k z n+1 = = Γ (αn + β) (n!)2 Γk (αn + α + β) [(n + 1)!]2 n=1 k n=0 γ+k (z) Wk,α,β

∞ n=0

−

γ Wk,α,β (z)

=

z zn zn (γ + k)n,k = k z Γk (αn + α + β) (n + 1)(n!)2 Γ (αn + α + β) (n + 1)(n!)2 n=0 k k γ

∞

γ(γ + k)n,k

Then, by Lemma 2 it results γ+k Wk,α,β (z)

−

γ Wk,α,β (z)

k d γ = (z) z Wk,α,β γ dz

In what follows shows the relationship between the k-Wrigth function and the k-Mittag-Leffler function obtained throung the Laplace transform. In fact, we have the following

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Theorem 1 Let α,β,γ∈ C, Re(α) > 0, Re(β) > 0, Re(γ) > 0, Re(s) > 0, s = 0. Then γ 1 γ L Wk,α,β (z) (s) = Ek,α,β (s−1 ) (II.7) s γ where Ek,α,β (z) is the k-Mittag-Leffler function given by Dorrego and Cerutti. (cf. [])

Proof. From definition of Laplace Transform and from (I.3) we have L

γ (z) (s) Wk,α,β ∞ n=0

+∞

= 0

−sz

e

∞ n=0

1 (γ)n,k Γk (αn + β) (n!)2

0

zn (γ)n,k dz = Γk (αn + β) (n!)2 +∞

e−sz z n dz

(II.8)

Taking into account that the integral in (II.8) is

+∞ 0

e−sz z n dz =

Γ(n + 1) n! = sn+1 sn+1

(II.9)

From (II.9) and (II.8) we have ∞ γ (z) (s) = L Wk,α,β n=0

1 s Then

II.2

∞ n=0

(s−1 ) (γ)n,k Γk (αn + β) (n!)

n+1

=

(s−1 ) 1 γ (γ)n,k = Ek,α,β (s−1 ) Γk (αn + β) (n!) s n

γ 1 γ L Wk,α,β (z) (s) = Ek,α,β (s−1 ) s

Fractional Calculus of the E (t, k, α, β) function

In this subsection we will defined the function E (t, k, α, β) in term of the kWrigth function and then we evaluate its Riemann-Liouville fractional integral and derivative. Definition 4 Let α, β, γ be complex numbers that Re(α) > 0, Re(β) > 0 and Re(γ) > 0, k > 0 and t ∈ R. We define the auxiliary function E (t, k, α, β) by the following relation β

α

γ E (t, k, α, β) = t k −1 Wk,α,β (t k )

(II.10)

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Proposition 1 Let α, β, γ, ν be complex number that Re(α) > 0, Re(β) > 0 and Re(ν) > 0, k > 0 and t ∈ R then β

α

γ (x k ) I+ν (E (t, k, α, β)) (x) = k ν x k +ν−1 Wk,α,β+kν

(II.11)

Proof. By definition (I.1) and (I.3) we have

I+ν

1 (E (t, k, α, β)) (x) = Γ(ν)

x

0

t

β −1 k

∞ n=0

1 (γ)n,k Γ(ν) n=0 Γk (αn + β) (n!)2 ∞

αn

tk (γ)n,k (x − t)ν−1 dt = Γk (αn + β) (n!)2 x

0

t

αn+β −1 k

(x − t)ν−1 dt

(II.12)

making the change of variable t = ξx, x − t = x(1 − ξ), dt = xdξ and replacing in (II.12) it result (γ)n,k 1 (E (t, k, α, β)) (x) = Γ(ν) n=0 Γk (αn + β) (n!)2 ∞

I+ν

αn+β 1 (γ)n,k x k +ε−1 2 Γ(ν) n=0 Γk (αn + β) (n!)

∞

0

1

ξ

1

(ξx)

αn+β −1 k

0

αn+β −1 k

[x(1 − ξ)]ν−1 xdξ =

(1 − ξ)ν−1 dξ

(II.13)

The integral in (II.13) result

1 αn+β αn + β −1 ν−1 ,ν ξ k (1 − ξ) dξ = B k 0 where B(z, w) is the Beta function. Then αn+β 1 (γ)n,k +ε−1 k (E (t, k, α, β)) (x) = x B Γ(ν) n=0 Γk (αn + β) (n!)2

∞

I+ν

αn + β ,ν k

αn+β ∞ Γ Γ(ν) αn+β 1 (γ)n,k k = x k +ε−1 αn+β 2 Γ(ν) n=0 Γk (αn + β) (n!) Γ k +ν

αn+β ∞ Γ(ν) k k −1 k ν Γ αn+β αn+β (γ)n,k 1 +ε−1 k x k αn+β+kν = αn+β+kν 2 Γ(ν) n=0 Γk (αn + β) (n!) k k Γ k

=

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α n ∞ (γ)n,k x k β 1 Γk (αn + β) Γ(ν) +ν−1 ν xk = k 2 Γ(ν) Γ (αn + β) (n!) Γk (αn + β + kν) n=0 k α β γ xk k ν x k +ν−1 Wk,α,β+kν

Proposition 2 Let α, β, γ, ν be complex number that Re(α) > 0, Re(β) > 0 and r − 1 0 and t ∈ R then α n ∞ xk (γ + k)n,k ν −ν αr+β −ν−1 D+ (E (t, k, α, β)) (x) = k x k (γ)r,k Γ (α(n + r) + β − kν) [(n + r)!]2 n=0 k (II.14) Proof. The Riemann-Liouvile fractional integral of order r − ν of E (t, k, α, β) result α β γ xk I+r−ν (E (t, k, α, β)) (x) = k r−ν x k +r−ν−1 Wk,α,β+(r−ν)k

(II.15)

Now we evaluate the derivative of order r with respect to x of (II.15) αn ∞ xk (γ)n,k dr r−ν β +r−ν−1 k k x = dxr Γ (αn + β + (r − ν)k) (n!)2 n=0 k

x (γ)n,k dr r−ν k r dx Γ (αn + β + (r − ν)k) n=0 k ∞

k

r−ν

(γ)n,k

αn+β k

n≥r

k r−ν

n≥r

αn+β +r−ν−1 k

(n!)2

=

αn+β + r − ν − 1 ... αn+β − ν − 1 x k −ν−1 k = Γk (αn + β + (r − ν)k) (n!)2

α

β

(γ)n,k Γ − 1 x k n x k −ν−1

= αn+β+(r−ν)k αn+β+(r−ν)k −1 2 Γ αn+β−kν − 1 k k Γ − 1 (n!) k k k

r−ν

n≥r

k

−ν

x

αn+β+(r−ν)k k

α

β

(γ)n,k x k n x k −ν−1 = αn+β−νk 2 k r k k −1 Γ αn+β−kν − 1 (n!) k β −ν−1 k

n≥r

α

(γ)n,k x k n = Γk (αn + β − kν) (n!)2

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k

−ν

x

β −ν−1 k

∞ n=0

α

α

(γ)n+r,k x k n x k r Γk (αn + αr + β − kν) [(n + r)!]2

(II.16)

and using the relation (γ)n+r,k = (γ)r,k (γ + k)n,k , (II.16) result

ν D+

(E (t, k, α, β)) (x) = k

−ν

x

αr+β −ν−1 k

(γ)r,k

∞ n=0

α n xk (γ + k)n,k Γk (α(n + r) + β − kν) [(n + r)!]2

Proposition 3 Let α, β, γ, ν be complex number that Re(α) > 0, Re(β) > 0, Re(γ) > 0 and j ∈ N. Then α n

j ∞ j(α−k)+β xk d (γ + k) n,k E (t, k, α, β) = k −j x k −1 (γ)j,k dt Γ (αn + β + j(α − k)) [(n + j)!]2 n=0 k (II.17) Proof. It is sufficient take ν = r = j in Proposition 2.

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[7] F. Mainardi. On the distinguised role of the Mittag-Leffler and Wrigth functions in fractional calculus. Special functions in the 21 st Century. Theory and Appl. Washington DC. USA. 6-8. April 2011. [8] F. Mainardi; G. Pagnini. The wrigth function as solutions of the timefractional diffusion equations. Appl. Math. and Computation. 141. 2003. [9] M. Mansour. Determinig the k-generalized Gamma Function Γ(x) by Functional Equations. Int. J. Contemp. Math. Science. Vol 4. No 21. 2009. [10] S. Mubeen; G. M. Habibullah. k-Fractional Integrals and Aplication. Int. J. Contemp. Math. Science. Vol. 7. N.2. 2012. Forthcoming paper. [11] I. Podlubny. Fractional Differential Equations. An introduction to fractional derivatives. Academic Press. 1999. [12] S. Samko; A. Kilbas; O. Marichev. Fractional integrals and derivatives. Gordon and Breach. 1993. Received: January, 2012