ABSTRACT

Let s and z be complex variables, be the Gamma function, and for any complex be the generalized Pochhammer symbol. Wright Type Hypergeometric Function is defined (Virchenko et al. [1]), as: where which is a direct generalization of classical Gauss Hypergeometric Function. The principal aim of this paper is to study the various properties of this Wright type hypergeometric function; which includes differentiation and integration, representation in terms of and in terms of Mellin-Barnes type integral. Euler (Beta) transforms, Laplace transform, Mellin transform, Whittaker transform have also been obtained; along with its relationship with Fox H-function and Wright hypergeometric function.

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.

REFERENCES