1. Introduction and Preliminaries
The Gauss Hypergeometric Function is defined [2] as:
; and (1.1)
The Generalized Hypergeometric Function, in a classical sense has been defined [3] by
(1.2)
and no denominator parameter equal to zero or negative integer.
E. Wright [4] has further extended the generalization of the hypergeometric series in the following form
(1.3)
where 
and 
are real positive numbers such that
When 
and 
are equal to 1, Equation (1.3) differs from the generalized hypergeometric function 
by a constant multiplier only.
The generalized form of the hypergeometric function has been investigated by Dotsenko [5], Malovichko [6] and one of the special cases considered by Dotsenko [5] as
(1.4)
and its integral representation expressed as
(1.5)
where
. This is the analogue of Euler’s formula for the Gauss’s hypergeometric function [3]. In 2001 Virchenko et al. [1] defined the said Wright Type Hypergeometric Function by taking 
in
(1.4) as
(1.6)
If
, then (1.3) reduces to a Gauss’s hypergeometric function. Galue et al. [7] and Virchenko et al. [1] investigated some properties of the function
.
The following well-known facts have been prepared for studying various properties of the function
.
• Euler (Beta) transform (Sneddon [8]):
The Euler transform of the function 
is defined as
(1.7)
• Laplace transform (Sneddon [8]):
The Laplace transform of the function 
is defined as
(1.8)
• Mellin transform (Sneddon [8]):
The Mellin transform of the function 
is defined as
(1.9)
then
(1.10)
• Wright generalized hypergeometric function (Srivastava and Manocha [9]), denoted by
, is defined as
(1.11)
(1.12)
where 
denotes the Fox H-function [10].
2. Basic Properties of the Function 
Theorem 2.1
(2.1.1)
(2.1.2)
In particular,
(2.1.3)
Proof.
which is the (2.1.1).
Now,
This is the proof of (2.1.2).
For 
and substituting 
in above result, this will immediately leads to particular case (2.1.3). 
Theorem 2.2 1) If
(2.2.1)
2) If
(2.2.2)
3)
(2.2.3)
In particular,
(2.2.4)
Proof.
1)
which concludes the proof of (2.2.1). 
2)
Therefore,
Which is the proof of (2.2.2). 
3)
This leads the proof of (2.2.3).
On putting
, in the above expression immediately leads to (2.2.4). 
Theorem 2.3
If 
(2.3.1)
Proof.
This establishes (2.3.1).
3. Representation of Wright Type
Hypergeometric Function 
in Terms of the Function 
Using the definition
, and taking
we have
(3.1)
where 
is a
-tuple
;
is a 
-tuple
.
Convergence criteria for generalized hyperfeometric function
:
1) If
, the function 
converges for all finite
.
2) If
, the function 
converges for 
and diverges for
.
3) If
, the function 
is divergent for
.
4) If
, the function 
is absolutely convergent on the circle 
if
.
4. Mellin-Barnes Integral Representation of 
Theorem 4.1 Let
Then 
is represented by the MellinBarnes integral
(4.1.1)
where
; the contour of integration beginning at 
and ending at
, and intended to separate the poles of the integrand at 
to the left and all the poles at 
as well as
to the right.
Proof. We shall use the sum of residues at the poles 
to obtain the integral of (4.1.1).
(4.1.2)
Now,
(4.1.3)
(4.1.2) and (4.1.3) completes the proof of (4.1.1). 
5. Integral Transforms of 
In this section we discussed some useful integral transforms like Euler transforms, Laplace transform, Mellin transform and Whittaker transform.
Theorem 5.1 (Euler (Beta) transforms).
(5.1.1)
Proof.
This is the proof of (5.1.1). 
Remark: Putting 
in (5.1.1), we get
(5.1.2)
Taking
, 
and substituting 
in place of the notation
; (5.1.1) reduces to
(5.1.3)
Also, considering 
and 
in (5.1.1), with replacement of 
by 
at
, we get
(5.1.4)
Theorem 5.2 (Laplace transform).
(5.2.1)
Proof.
This is the proof of (5.2.1). 
Theorem 5.3 (Mellin transform).
(5.3.1)
Proof. Putting 
in (4.1.1), we get
(5.3.2)
where, 
Using (1.9), (1.10), and (5.3.2) immediately lead to (5.3.1). 
Theorem 5.4 (Whittaker transform).
(5.4.1)
Proof. To obtain Whittaker transform, we use the following integral:
where 
Substituting 
on the L.H.S. of (5.4.1), it reduces to
This completes the proof of (5.4.1).
6. Relationship with Some Known Special Functions (Fox H-Function, Wright Hypergeometric Function)
6.1. Relationship with Fox H-Function
Using (4.1.1), we get
6.2. Relationship with Wright Hypergeometric Function
The Generalized Hypergeometric Function 
as in (1.3) is
(6.2.1)
From (1.11) and (6.2.1) yields
. (6.2.2)
7. Acknowledgements
The authors are thankful to the reviewers for their valuable suggestions to improve the quality of paper.