On the Fourier Transform of the Products of M-Wright Functions

and state the Fourier transform of this function. Moreover, using the new integral representations for the products of the M-Wright functions, we also get the Fourier transform of it.


Introduction and Preliminaries
The M-Wright function or the Mainardi function [13] with the well-known differential equation for α = 1 ν , plays an important role in fractional calculus.This function is derived from the Wright function A. Ansari and for the first time was introduced by Mainardi at the beginning of the 1990's for expressing the fractional diffusion-wave equations.Later, other researchers stated some prominent roles of this function in analyzing partial fractional differential equations and statistical distributions.Most of these works have been surveyed using the operational calculus of this function such as the Laplace and Mellin transforms.See for example [3,4,5,6], [9], [11,12] and [15].Therefore for the importance and significance of this function, in this paper, we get the Fourier transform of the M-Wright and its associate functions.For this purpose, using the Laplace integral we derive an integral representation for the special case M 1 2n+1 (x).Then, by applying the Bromwich's integral for the inverse Mellin transform we find an extended class of integrals representation for the M-Wright function.Finally, we obtain the Fourier transform of the M-Wright function and the products of M-Wright functions.First, we consider the following ordinary differential equation of order 2n which has 2n linear independent solutions in terms of the generalized hypergeometric functions [10] y One of the solutions can be obtained using the Laplace integral method By inserting the relation (1.5) in (1.4), we get a first order differential equation and find the solution y(x) as the following integral representation (except for a normalization constant) The contour C is chosen such that the function v(z) must vanish at boundaries.For this purpose, the real part of z 2n+1 must be positive and can be considered as paths between two lines of set {re or equivalently (1.9) On The Fourier Transform of The Products of M-Wright Functions 247

Main Theorems
In this section, we establish some theorems and corollaries on the operator e −λΦ(s) .First, we derive an integral representation for the operator e −λΦ(s) .Theorem 2.1.Let Φ(s) be an entire function such that for even and odd functions u(r) and v(r), we have φ(c±ir) = h(c)(u(r)±iv(r)) where h is an analytic function.Then for c 1 < ℜ(s) < c 2 the following relation holds true where the function A(ξ, λ) is presented by Proof: By definition of the inverse Mellin transform of function e −λΦ(s) and transition of the contour integration ℜ(s) = c, we have ∞ 0 e −λu(r) e +iλv(r)+ir ln(t) dr r) cos(r ln(t) + λv(r))dr. (2. 3) The relation (2.3) implies that the Mellin transform of the last integral is equal to function e −λΦ(s) , that is which by setting ln(t) = ξ, we get the relation (2.1).✷

A. Ansari
Corollary 2.2.If we suppose that the function Φ(ir) can be written as the real function Φ 1 (ir) = u(r) or pure imaginary function Φ 2 (ir) = iv(r), then, the relation (2.1) can be changed into two special cases: where the functions A K and A I are given by where the A 2n+1 (ξ) function is given by ))dr. (2.9) For more details about this function and its asymptotic expansion at infinity, see [2], [14].
(2.10) Theorem 2.5.(The Schouten-Vanderpol theorem for the Mellin transform) Let F (s) and φ(s) be analytic functions in the strip c 1 < ℜ(s) < c 2 and let F (s) be the Mellin transform of f (t), then the inverse Mellin transform of F (φ(s)) is written by

11)
Proof: Using the definition of Mellin transform of F (φ(s)) On The Fourier Transform of The Products of M-Wright Functions 249 and by replacing in the inverse Mellin transform of F (φ(s)) we have Now, by changing the order of integration, we arrive at the relation (2.11).✷ Corollary 2.6.It is obvious that by setting φ(s) = s α , 0 < α < 1, in the relation (2.11) and using the following Bromwich's integral in terms of the M-Wright function [1] the inverse Mellin transform of F (s α ) can be presented in terms of the M-Wright function as follows Corollary 2.7.By setting F (s) = e −λΦ(s) in the relation (2.1) and combining with (2.13), we get a new integral representation for the fractional exponential operator e −λΦ(s α ) as follows where the function A α (ξ, λ) is given by (2.15)

The Fourier Transform of The M-Wright Function
In this section, using the obtained relations in previous section we get the Fourier transform of the M-Wright function and its associate functions.First, we get an integral representation for the products of A 2n+1 (x) functions.
Lemma 3.1.The following identity holds for the product of where the A * 2n+1 (x, z) function in two variables is given by

.2)
A. Ansari Proof: With the help of the integral representation (1.9), we consider the following product of the A 2n+1 (x) functions dxdy. (3.3) Now, using the following change of variables the integral (3.3) leads to the inverse Fourier transform of the A * 2n+1 (x, z) function in two variables 2), we can derive some special cases iii On The Fourier Transform of The Products of M-Wright Functions 251 iv) u = 0 Proof: By the identity (3.6), we take 2n derivatives of the function A 2n+1 (u)A 2n+1 (−u) as follows )w 2n dw. (3.11) The above relation can be expressed as the Fourier transform of the following function which implies that 2n− 1 first moments of ); w} = (2n + 1) F{A α (ξ, λ); w} = e −λΦ((−iw) α ) , (3.16) ).
(3.17 Proof: Using the definition of the M 1 2n+1 (ξ) function in (2.8) ) and applying the following fact for the Dirac delta function we get the desired result 2n+1 -transform) with its inversion formula as

Concluding Remarks
This paper provides new results on the operational calculus of the M-Wright function.Theses results have been obtained using the new integral representations of the exponential function in some special cases.These integral representations led to the Fourier transform of the M-Wright function and its associate functions.Also, we found the Fourier transform of the products of the M-Wright functions according to the relation (3.6) and we saw that this product is orthogonal according to the relation (3.18).Also, we showed that 2n − 1 first moments of the function 2kπi 2n+1 , r > 0, k = 0, 1, • • • , 2n}.Anyway, after the deformation and normalization of integral (1.7), we rewrite the function y as the A 2n+1 (x) function (the generalized airy function) as follows[2]

Theorem 3 . 4 .
The following relation have been held for the Fourier transform of A. Ansari the M-Wright function and its associate functions F{A(ξ, λ); w} = e −λΦ(−iw) ,

✷Corollary 3 . 6 .
On The Fourier Transform of The Products of M-Wright Functions 253 Using the relation (3.18) for the orthogonality condition of the M 1 2n+1 (ξ) function, we can get a new integral transform (which we denote as the M 1