Abstract
The paper is devoted to the study of the function , which is an extension of the classical Wright function and Kummer confluent hypergeometric function. The properties of including its auxiliary functions and the integral representations are proven.
1. Introduction
The special functions of mathematical physics are found to be very useful for finding solutions of initial and/or boundary-value problems governed by partial differential equations and fractional differential equations. Special functions have widespread applications in other areas of mathematics and often new perspectives in special functions are motivated by such connections. Several special functions, called recently special functions of fractional calculus, play a very important and interesting role as solutions of fractional order differential equations, such as the Mittag-Leffler function, Wright function with its auxiliary functions, and Fox’s -function.
The Wright function is one of the special functions which plays an important role in the solution of linear partial fractional differential equations. It was introduced for the first time in [1, 2] in connection with a problem in the number theory regarding the asymptotic of the number of some special partitions of the natural numbers. Recently this function has appeared in papers related to partial differential equations of fractional order. Considering the boundary-value problems for the fractional diffusion-wave equation, that is, the linear partial integrodifferential equation obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order with , it was found that the corresponding Green functions can be represented in terms of the Wright function. Furthermore, extending the methods of Lie groups in partial differential equations to the partial differential equations of fractional order, it was shown that some of the group-invariant solutions of these equations can be given in terms of the Wright and the generalized Wright functions [3]. A list of formulas concerning this function can be found in the handbook of Bateman Project, Erdélyi et al. 1953 [4]; see also [3, 5–11].
The Wright function is defined by the series representation, valid in the whole complex plane It is an entire function of order , which has been known also as generalized Bessel (or Bessel Maitland) function [12, 13]. There are two auxiliary functions of Wright function defined aswhere the function is recently known as the Mainardi function [12, 13]. In a continuation of this study, we investigate the generalized Wright function which is defined for real and with and with as where is a Pochhammer symbol and is a gamma function. The function is an entire function of order and generalized Wright function (1).
By analogy with (2), one can also introduce the two (generalized Wright type) auxiliary functions for any order and for all complex variable by
2. Basic Definitions
In this section, the definitions of some special functions and their properties necessary for our studies on generalized Wright function (3) and its auxiliary functions (4) and (5) will be presented to provide convenient references.
2.1. Wright Function
The Wright function which is defined by (1) has the Hankel contour integral representation and special casewhere is the Hankel contour and is the complementary error function which is defined as The important relation between two auxiliary functions (2) of Wright function (1) and some special cases arewhere is Airy’s function which is the one of the linearly independent solutions of the ordinary differential equation and has the following properties [14, 15]:Airy’s function has, from its differential equation, the following indefinite integrals:
2.2. Mittag-Leffler Function
A two-parameter function of the Mittag-Leffler type is defined by the series expansion [12, 13] It follows from the definition that Let be an integer, let , and let be arbitrary real numbers. By means of “multi-index” , we introduce the so-called multi-index Mittag-Leffler functions [16, 17]:
2.3. Fox-Wright Function
The Fox-Wright function has series representation [5, 12, 18–20]: The Fox-Wright -function is special case of Fox’s -function: where denotes the Fox -function [21] which can be expressed as Mellin-Barnes contour integral representation in the formwhere is a suitable contour in the complex -plane, the orders are integers , , and the parameters , , , and , , , are such that .
2.4. Meijer -Function
In 1936, C. S. Meijer introduced the -function as [14]where , , and is the generalized hypergeometric function which is defined as If , we have the confluent hypergeometric function which has the integral representation with the property
2.5. Gamma Function and Incomplete Gamma Function
The contour integral representation for the reciprocal gamma function, multiplication formula, and important property [22] areThe upper and lower incomplete gamma function, respectively, areThe two types of incomplete gamma function have the properties
3. Integral Representations of the Generalized Wright Function
Here, we introduce the Mellin-Barnes contour integral representation and definite integral representation of function (3).
Theorem 1 (Mellin-Barnes contour integral representation). Let , and Then function (3) can be represented by the Mellin-Barnes contour integral as where is not equal to zero and in which denotes the natural logarithm and is not necessarily the principal value, and is a suitable path in the complex -plane which runs from to , so the points , , lie to the right of a contour and the points , , lie to its left.
Proof. Consider the integral in (31) with the contour replaced by the contour consisting of a large clockwise-oriented semicircle of radius and the center of the origin which lies to the right of the contour and is bounded away from the poles. We can apply Cauchy’s theorem (Residues Theorem) to the closed contour which is consisting of the contour and that part of terminated above and below by as ; we obtain
Theorem 2. Let , and . Then function (3) can be represented as and consequently one can get
Proof. Using the contour integral representation for reciprocal gamma function (25), the integral representation of confluent hypergeometric function (23), and the Hankel contour integral representation of Wright function (6), then
4. Relationship with Some Known Special Functions
We devote this section to studying the relationship between generalized Wright function (3) and some known special functions like Fox -function, Fox-Wright function, Meijer -function, Mittag-Leffler function, and generalized hypergeometric function.
4.1. Relation with Fox -Function
Using (31) and the definition of Fox -function, we obtain
4.2. Relation with Fox Wright Function
Generalized Wright function (3) can be represented by Fox Wright function (19) as
4.3. Relation with Meijer -Function
From the definitions of generalized Wright function (3) and Meijer -function (22) with , we obtain
Theorem 3. Let be a rational number with and ; then one hasIn particular, if ,
Proof. Using the contour integral representation of generalized Wright function (31) with change of the variable from to and multiplication formula (26), we get
4.4. Relation with Mittag-Leffler Function
Let ; from definition (3), we have where is Mittag-Leffler function (16). If we put with using formula (17), then we obtain In the same way, if we put and , then where is the multiple Mittag-Leffler function which is defined by formula (18).
4.5. Relation with
Theorem 4. Let , where ; then one has
Proof. From the definitions of generalized Wright function (3) and confluent hypergeometric function ((23); (24)), we have
Theorem 5. Let , where ; then one has
Proof. From the definitions of generalized Wright function (3) and generalized hypergeometric function, we have
Theorem 6. Let with , and , where is not integer number; thenand, in particular, if ,
Proof. From the definitions of generalized Wright function (3) and formula (27), we have
Theorem 7. Let ; , and ; then one can deduce that
Proof. From the integral representation of generalized Wright function (34) and formula (7), we can deduce that
Theorem 8. Let , and ; then one can deduce that
Proof. From the integral representation of the auxiliary functions of generalized Wright function (35) and formula (10), we can deduce that
Theorem 9. Let , and ; then one can deduce thatIn particular, if , Finally, if ,
Proof. From the integral representation of the auxiliary functions of generalized Wright function (35) and formula (10), we can deduce that
5. Basic Properties of the Generalized Wright Function
Theorem 10. The important relation between the two auxiliary functions ((4); (5)) of generalized Wright function (3) is
Proof. By using the integral representations of two auxiliary functions ((35); (36)) and formula (9), we have
Theorem 11. Let , , and ; and ; then one can arrive at where is incomplete gamma function (29) and is error function (8).
Proof. From the integral representation of generalized Wright function (34) and formula (7), we have
Remark 12. In the case , , we have and in the case , , we have
Theorem 13. Let , ; and ; then one has where is the incomplete gamma function which is defined by (29).
Proof. From integral representation (35) of the auxiliary function with using relation (10) and changing the variable , we have
Corollary 14. Let , , be an odd integer; one has and if , , is an even integer, one has
Theorem 15. Let and ; then first auxiliary function (4) of the generalized Wright function has the forms
Proof. From the integral representation of (35) and relation (11), we can deduce that In the above relation if we change the variable and using the relations ((14)-(15)) and integral of Airy’s function with putting , , and , , then the proof is completed.
6. Recurrence Relations
Theorem 16. Let ; ; , ; and , then one can deduce the following recurrence relation: and, in particular, if ,
Proof. Consider
Theorem 17. Let ; ; , , and ; then one can deduce that
Remark 18. The above theorem can be returned to Theorem 10 if we substitute , and .
7. Integral Transforms of the Generalized Wright Function
In this section, we will introduce the Euler transform, Laplace transform, and Mellin transform of the function .
7.1. Euler Transform
The Euler transform of the function follows from the beta function
7.2. Laplace Transform
Consider
7.3. Mellin Transform
Consider
8. Derivative of the Generalized Wright Function
Theorem 19. Let ; and ; then the first derivative of generalized Wright function (3) has the form And the higher derivative is
Proof. From definition (3), we have By repeating this process -times, we arrive at the second requisition.
Theorem 20. Let ; and , , and ; then the derivative of auxiliary function (4) of generalized Wright function (3) has the form and, for at , one obtains
Proof. From integral representation (35), we haveand one can use the relation between two auxiliary functions of generalized Wright function (59) to complete the proof of the first requisition. To obtain the second requisition, we start with definition (4) and reflection formula of gamma function, as when ; then we get
Theorem 21. Let ; , , , and ; then one has In particular, if , then one obtains
Proof. Consider
9. Conclusion
In this paper, we generalize the definition of Wright function (1) and its auxiliary functions (2) to be the function defined as in (3) and its auxiliary functions (4) and (5). The properties of including its auxiliary functions and the integral representations are provided and proven. The relationship with some known special functions like Fox -function, Fox-Wright function, Meijer -function, Mittag-Leffler function, and generalized hypergeometric function are given. The Euler transform, Laplace transform, and Mellin transform of function (3) are introduced.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.