A new generalized Wright function and its properties
aa r X i v : . [ m a t h . G M ] F e b A new generalized Wright function and its properties
R. DrogheiMinistero dell’Istruzione, dell’Universit`a e della Ricerca,IIS-Ceccano, Via Gaeta, 105, 03023 Ceccano (FR), Italy.E-Mail Address: [email protected]
Abstract
In this paper we introduce a new multiple-parameters generalization of the Wrightfunction arose from an eigenvalues problem concerning an hyper-Laguerre-type op-erator involving Caputo derivatives. We show that by giving particular values at theparameters including in the function, it leads right to well known special functions (theclassical Wright function, the α -Mittag-Leffler function, the Tricomi function etc...). Inaddition, we investigate a nonlinear fractional differential equation admitting the newgeneralization Wright function as solution, and in particular isochronous solutions. Keywords: Special Function of Fractional Calculus, Wright function, Caputo deriva-tives, nonlinear fractional PDEs, Laguerre derivatives operator.
Special functions play an essential role in every fields of mathematical physics. In fact, insolving several problems of these ambits one is led to use various special functions, becauseany analytical solutions are expressed in terms of some of these functions. Applied scien-tists and engineers, dealing with practical application of differential equations, see the roleof special functions as an important mathematical tool. Recently, there has been an in-creasing interest to use special functions in mathematical models involving fractional order of differential equation and systems to investigate various physical, biological, biomedical,chemical, economical etc, phenomena. For this we refer the reader to the books [1, 2, 3] andreferences therein. These special functions, such as: Mittag-Leffler function, Wright func-tions with its auxiliary functions and Fox’s H -functions, precisely for this reason, are calledSF of FC (Special Function of Fractional Calculus)(see [4, 5]). The Wright function is oneof the SF of FC which plays a prominent role in the solution of linear partial fractional dif-ferential equation. In particular, one decades ago, the Wright function appeared in articles1elated to partial differential equation of fractional order: boundary-value problems for thefractional diffusion-wave equation [12]. Otherwise, it was introduced and investigated in aseries of number theory notes from 1933 ([9, 10, 11]) in the framework of the theory of par-titions. In recent times, several generalizations of the Wright function have been proposed:the M-Wright function entering a relevant class of self-similar stochastic processes thatwe generally refer as time-fractional diffusion processes [13]; a four parameters extensionof Wright function W γ,δα,β ( z ) was introduced and studied in [14]. In this paper we solve afractional differential equation involving Laguerre-derivative-type operator. An importantproperty of the solution found is that in specific cases is possible to refer to well known SFof FC (Laguerre-exponential function, n-Mittag-Leffler function, classical Wright functionsand Tricomi function).The paper is organized as follow. In section 2, we introduce a fractional differential equa-tion involving a fractional hyper-Laguerre-type operator. In particular, we solve a fractionaldifferential equation involving the previously defined operator. We found that the solu-tion, called successively the multiple-parameters generalized Wright functions ( W (¯ α, ¯ ν ) ( z )),is defined by series representation and is an entire in the complex plane. In section 3, we in-vestigate particular assignments of the parameters included in the W (¯ α, ¯ ν ) ( z ), in such a wayto retrieve well known SF of FC. In section 3, we study a particular fractional isochronouspartial differential equation involving Caputo derivative in space and admitting explicit,completely periodic, solution in separating variable form. Considering the well known
Laguerre derivative operator investigated in [8], defined as D nL = ddx x . . . ddx x ddx x ddx | {z } n +1 derivatives ; (1)and successively analysed in fractional version, namely hyper-Bessel-type operator, byGarra and Polito in [6] d ν dx ν x ν . . . d ν dx ν x ν d ν dx ν x ν d ν dx ν | {z } n +1 derivatives ; (2)with x ≥ , ν > d ν dx ν representing the Caputo fractional derivative [17, 18].We introduce a fractional hyper-Laguerre-type operator defined as D (¯ α, ¯ ν ) nL = d α n +1 dx α n +1 x ν n d α n dx α n x ν n − d α n − dx α n − · · · x ν d α dx α ; (3)where ¯ α = ( α , ..., α n +1 ); ¯ ν = ( ν , ..., ν n ) and d αj dx αj , j = 1 , ..., n +1 are Caputo fractionalderivatives and α j > , j = 1 , ..., n + 1 and ν j > , j = 1 , ..., n .2 efinition 2.1 ( m-p generalized Wright function ) . A multiple parameters generalizedversion of the Wright function W (¯ α, ¯ ν ) ( z ) is defined by the series representation as a functionof the complex argument z and the parameters α j , j = 1 , ..., n + 1 and ν j , j = 1 , ..., n W (¯ α, ¯ ν ) ( z ) = ∞ X k =0 k Y i =1 n Y j =1 Γ( α n +1 i + a j )Γ( α n +1 i + b j ) · z k Γ( α n +1 k + b n +1 ) ; (4)where a j = 1 + j X m =1 ( ν m − − α m ) , b j = 1 + j X m =1 ( ν m − − α m − ) . The W (¯ α, ¯ ν ) ( z ) is an entire function for α j > , j = 1 ..n + 1; ν j ∈ C , j = 1 ..n and α = ν = 0. Theorem 2.1.
The multiple parameters generalized Wright function W (¯ α, ¯ ν ) ( λx α n +1 ) with λ ∈ R , x ≥ , α j > , j = 1 , ..., n + 1 and ν j > , j = 1 ..n satisfies the following fractionaldifferential equation involving fractional hyper-Laguerre-type operator (3) D (¯ α, ¯ ν ) nL W (¯ α, ¯ ν ) ( λx α n +1 ) = λx P ns =1 ( ν s − α s ) W (¯ α, ¯ ν ) ( λx α n +1 ) . (5) Proof.
Applying recursively ( n + 1) − times the Caputo derivatives on the function, wefind the following expression (see the FC’s properties in Appendix B) d α n +1 dx α n +1 x ν n d α n dx α n x ν n − d α n − dx α n − · · · x ν d α dx α W (¯ α, ¯ ν ) ( λx α n +1 )= d α n +1 dx α n +1 ∞ X k =1 k Y i =1 n Y j =1 Γ( α n +1 i + a j )Γ( α n +1 i + b j )Γ( α n +1 k + b n +1 ) n Y m =1 Γ( α n +1 k + b m )Γ( α n +1 k + a m ) λ k x α n +1 k − α ···− α n + ν ··· + ν n = ∞ X k =1 k Y i =1 n Y j =1 Γ( α n +1 i + a j )Γ( α n +1 i + b j )Γ( α n +1 k + b n +1 ) n Y m =1 Γ( α n +1 k + b m )Γ( α n +1 k + a m ) Γ( α n +1 k + b n +1 )Γ( α n +1 k + a n +1 ) λ k x α n +1 k + P ns =1 ( ν s − α s ) = λx P ns =1 ( ν s − α s ) ∞ X k =1 k − Y i =1 n Y j =1 Γ( α n +1 i + a j )Γ( α n +1 i + b j ) λ k − x α n +1 ( k − Γ( α n +1 ( k −
1) + b n +1 )= λx P ns =1 ( ν s − α s ) ∞ X k =0 k Y i =1 n Y j =1 Γ( α n +1 i + a j )Γ( α n +1 i + b j ) λ k x α n +1 k Γ( α n +1 k + b n +1 )= λx P ns =1 ( ν s − α s ) W (¯ α, ¯ ν ) ( λx α n +1 ) . Lemma 2.2.
In case of P ns =1 ( ν s − α s ) = 0 , the multiple parameters generalized Wrightfunction W (¯ α, ¯ ν ) ( λx α n +1 ) becomes eigenfunction of the fractional hyper-Laguerre-type oper-ator (3). Particular cases
In this section, we will assign particular values at the parameters of the W (¯ α, ¯ ν ) ( z ) in sucha way that we can identify it with special functions well known in literature. Remark . Laguerre-exponential function
Now we want to consider the particular, integer order, case in which the α j = ν j = 1 , j =1 , · · · , n ; α n +1 = 1 and therefore a j = 0; b j = 1 , j = 1 , · · · , n + 1.It is simply to verify that multiple parameters Wright function matches Laguerre-exponential function e n ( x ) = ∞ X k =0 x k ( k !) n +1 (6)investigated in [8]. Remark . n-Mittag-Leffler function Similarly, in case of α i = ν i = 1 , ..., n + 1 and ν i = ν i = 1 , ..., n the multiple parametersWright function (4) becomes the n-Mittag-Leffler function, with n ∈ N − { } E n ; ν, ( x ) = ∞ X k =0 x k Γ n +1 ( νk + 1) , x ≥ , ν > . (7)Moreover, the hyper-Laguerre-type operator in (5) assumes the hyper-Bessel-type operatorform investigated in [6]. Definition 3.1.
We analyse the particular case n = 1 with α = β, α = α and ν = ν W α,β,ν ( x β ) = ∞ X k =0 k Y i =1 Γ( βi + 1 − α )Γ( βi + 1) x βk Γ( βk + 1 − α + ν ). Proposition 3.1.
Obviously, this particular case 3.1 satisfies the following fractional dif-ferential equation d β dx β (cid:18) x ν d α dx α f ( x ) (cid:19) = x ν − α f ( x ) , (8) involving two fractional derivatives in the sense of Caputo of orders α, β ∈ (0 , . Where f ( x ) = W α,β,ν ( x β ) .Remark . Classical Wright function
In the case α = 1 the function W ,β,ν ( βx β ) (definition 3.1) coincides with the classicalWright function W β,ν ( x β ). W ,β,ν ( βx β ) = ∞ X k =0 k !Γ( βk − ν ) x βk = W β,ν ( x β )4 emark . Tricomi function
For α = β = ν = 1 the function W , , ( x ) matches the Tricomi function C ( x ) = ∞ X k =0 x k ( k !) ;that is an eigenfunction of the Laguerre derivative D L , (see [15]) and it is directly relatedto the Bessel function. Recently, SF of FC are used used to solve space or time-fractional version of ω -modified PDEs ( isochronous PDEs ) admitting explicit solution (see [16]). As a simple application,we analyse a nonlinear fractional PDE with the remarkable property to have isochronoussolution, i.e. completely periodic solutions with fixed period T = πω . Considering theproposition (3.1), the following equation ∂u ( x, t ) ∂t + iωu ( x, t ) = ∂ β ∂x β x ν ∂ α u ( x, t ) ∂x α + ikx ν − α u ( x, t ); (9)has an explicit separable variables isochronous solution: u ( x, t ) = exp( iωt ) · W α,β,γ ( − ikx β ) (10) As mentioned above, we deem that the main result of this paper is related to the functionobtained by solving the fractional differential equation (5) involving a fractional hyper-Laguerre-type operator (3). We have not found such function in literature, but of course wecannot be quite certain that such function is new. We suppose that it could be a particularcase of a more general special function. It seem to us in any case remarkable that suchresult can be obtained- presumably for the first time- from a solution of a fractional ODE.
Appendix
Appendix A. Convergence radius of the multiple parameters generalizedWright function
Using the Wendel’s asymptotic formula [7]Γ( s + a )Γ( s + b ) = s a − b (cid:2) O ( s − ) (cid:3) , | s | → ∞ , | arg ( s ) | < π. (11)5e compute the convergence radius of the multiple parameters generalized Wrightfunction using the D’Alembert Criteria (Ratio Test).lim k → + ∞ (cid:12)(cid:12)(cid:12)(cid:12) a k +1 a k (cid:12)(cid:12)(cid:12)(cid:12) = lim k → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ( α n +1 k + b n +1 )Γ( α n +1 ( k + 1) + b n +1 ) n Y j =1 Γ( α n +1 ( k + 1) + a j )Γ( α n +1 ( k + 1) + b j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =lim k → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( α n +1 k ) − α n +1 n Y j =1 ( α n +1 k ) − α j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =lim k → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n +1 Y j =1 ( α n +1 k ) − α j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 (12)considering α j > , j = 0 ..n + 1.It means that the multiple parameters generalized Wright function has infinite radiusof convergence. Appendix B. Fractional calculus and Caputo derivatives
In order to make the paper self-contained, we briefly recall main definitions and propertiesof fractional calculus operators.Let γ ∈ R + . The Riemann-Liouville fractional integral is defined by J γx f ( t ) = 1Γ( γ ) Z x ( x − x ′ ) γ − f ( x ′ ) dx ′ , (13)where Γ( γ ) = Z + ∞ x γ − e − x dx, is the Euler Gamma function.Note that, by definition, J x f ( t ) = f ( t ).Moreover it satisfies the semigroup property, i.e. J αx J βx f ( t ) = J α + βx f ( t ).There are different definitions of fractional derivative (see e.g. [18]). In this paper we usedthe fractional derivatives in the sense of Caputo [17], that is D γx f ( x ) = J m − γx D mx f ( x ) = 1Γ( m − γ ) Z x ( x − x ′ ) m − γ − d m d ( x ′ ) m f ( x ′ ) d x ′ , γ = m. (14)6t is simple to prove the following properties of fractional derivatives and integrals (seee.g. [18]) that we used in the paper: D γx J γx f ( x ) = f ( x ) , γ > , (15) J γx D γx f ( x ) = f ( x ) − m − X k =0 f ( k ) (0) x k k ! , γ > , x > , (16) J γx x δ = Γ( δ + 1)Γ( δ + γ + 1) x δ + γ γ > , δ > − , t > , (17) D γx x δ = Γ( δ + 1)Γ( δ − γ + 1) x δ − γ γ > , δ > − , t > . (18) Acknowledgement
The author is grateful to Dr. Roberto Garra for providing essential information.