Research Article | Open Access
Moustafa El-Shahed, Ahmed Salem, "-Analogue of Wright Function", Abstract and Applied Analysis, vol. 2008, Article ID 962849, 11 pages, 2008. https://doi.org/10.1155/2008/962849
-Analogue of Wright Function
We introduce a -analogues of Wright function and its auxiliary functions as Barnes integral representations and series expansion. The relations between -analogues of Wright function and some other functions are investigated.
In the second half of the past century, the discoveries of new special functions and applications of special functions to new areas of mathematics have initiated a resurgence of interest in this field . These discoveries include work in combinatories initiated by Schutzenberger and Foata. Moreover, in recent years, particular cases of long familiar special functions have been clearly defined and applied to orthogonal polynomials .
The Wright function is one of the special functions, which plays an important role in the solution of fractional differential equations. The Wright function is defined by the series representation for all complex variable as :The Barnes integral representation of Wright function is defined bywhere is a contour in the complex -plane which runs from to , so that the points lie to the right of . There are two auxiliary functions of Wright function defined asHere, we listed some of special cases of Wright function and its auxiliary functions :where Erfc is the complementary error function, are the Bessel and modified Bessel functions, and is Airy's function, defined as
2.1. -Hypergeometric Series
A -hypergeometric series is a power series in one complex variable with power series coefficients which depend, apart from , on complex upper parameters and complex lower parameters as follows :where when .
2.2. -Exponential Series
2.3. Jackson -Derivative
In 1908, Jackson reintroduced and started a systematic study of the -difference operator [1, 3]:which is now sometimes referred to as Euler-Jackson , Jackson -difference operator, or simply the -derivative. By definition, the limit as -approaches to is the ordinary derivative, that is,if is differentiable at . The two exponential functions have the -derivative:
2.4. Jackson -Integrals
Thomae (1869) and Jackson (1910) introduced the -integral defined in Jackson also defined an integral from to byprovided the sums converge absolutely.
The -Jackson integral in a generic interval is given byThe -integration by parts is given for suitable functions and by 
2.5. -Gamma Function
Jackson (1910) defined the -analogue of the gamma function byMoreover, it has the -integral representations [1, 3]:Let denote a positive integer. In the same spirit we defineIt is obvious from (2.10) that has simple poles at . The residue at is A -analogue of Legendre's duplication formula can be easily derived bySimilarly, it can be shown that Gauss multiplication formula has a -analogue of the form
Lemma 2.1. Let be a positive integer and a nonnegative integer, then one has
The proof of this lemma follows from the definition of -analogue of gamma function.
2.6. Jackson's -Bessel Function
Jackson introduced in 1905 the following -analogues of the Bessel functions :
3. The -Analogue of Error Functions
Definition 3.1. One defines the -analogues of error function and complementary error function, respectively, as
Remark 3.2. If or in the above definitions, respectively, then we haveand we can deduce that
Proof. The series representations of the -error function are as follows:
4. -Analogue of Wright Function
In this section, we introduce a definition of a -analogue of Wright function () as a Barnes integral representations.
Definition 4.1. According to standard notation, one defines a -analogue of Wright function aswhere , , is not equal to zero andwhere is a suitable path in the complex -plane that runs from to , so the points , lie to the right of the contour .
4.1. Existence and Representation of -Wright Function
Firstly, we rewrite the definition of -Wright function asNext, we consider a -analogue of Wright function in the case that . Let ; using the triangle inequality, we getwhich is bounded on the contour .
Theorem 4.2. Let be a positive integer, and let be a complex number, then the -Wright function is absolutely convergent for all complex variables ; if and if , then .
Proof. Consider the integral in (4.1) 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 is bounded away from the poles.
Let , then we havewhereas on , it follows that from (4.5) the integral (4.1) with replaced by tends to zero as if and only if for all complex variable and if , then .
Theorem 4.3 (explicit power series expansion). Let be a positive integer, let be a complex number, and let either and or and . Then the -Wright function (4.1) has the power series expansion
Proof. From the existence theorem after replacing by , 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 that the -analogue of Wright function (4.1) equals the sum of the residues of the integrand at . This completes the proof.
Definition 4.5 (the auxiliary functions of -Wright function). We introduce two (-Wright-type) auxiliary functions and with . The two functions can be define for any order and for all complex variable by
Remark 4.6. An important relationship between auxiliary functions of -Wright function aswhen , we get
5. Relation with Some Known Special Functions
It follows from the definition of the -analogue of Wright function as a series expansion (4.7) thatThe Jackson's third -Bessel function and modified third -Bessel function can be expressed in terms of -Wright function asThe -error function complement can also be expressed a particular case of -Wright function aswhere denotes the -error function complement which is defined as in (3.2). To prove this formula, we use the definition of -Wright function (4.7) and the identities of the -gamma function:Taking and in the definition of -Wright function, then we obtainExplicit expressions of and in terms of known functions are expected for some particular values of . In the particular case , we findTo prove the first formula and the second formula, we use the relationship between them (4.11). Using the definition of -auxiliary Wright function (4.8) and the identities of the -gamma function (2.16), we obtain the following:In the case of , we can deduce thatwhere is the -analogue of the Airy function which is defined aswhere is Jackson's modified second -Bessel. To prove this formula, we use the definition of -Wright function and the identities of the gamma function (2.16):when , we cover the classical results about the Wright function and its auxiliary functions.
- G. E. Andrews, R. Askey, and R. Roy, Special Functions, vol. 71 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1999.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
- G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 2nd edition, 2004.
Copyright © 2008 Moustafa El-Shahed and Ahmed Salem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.