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Recently, an extended operator of fractional derivative related to a generalized Beta function was used in order to obtain some generating relations involving the extended hypergeometric functions [

For the sake of clarity and easy readability, we find it to be natural and convenient to divide this introductory section into three parts (or subsections). In Part 1.1, we introduce the extended Beta, Gamma and hypergeometric functions. Part 1.2 deals with the familiar Riemann–Liouville fractional derivative operator and its generalizations, which are motivated essentially by the definition in Part 1.1 for the extended Beta function. In the third subsection (Part 1.3), we then introduce the extended Appell hypergeometric functions in two variables, which were recently investigated in conjunction with the family of the extended Riemann–Liouville fractional derivative operators defined in Part 1.2.

Extensions of a number of well-known special functions were investigated recently by several authors (see, for example [

so that, clearly,

Here, and in what follows, such arguments as (for example)

Making use of the extended Beta function

where

it being understood

Among several interesting and potentially useful properties of the extended hypergeoemetric function

Obviously, for the Gauss hypergeometric function

The following

and

respectively,

Clearly, since

the special cases of Equations (1.6–1.8) when

Throughout our present investigation, it is

For the

it is known that

where, and in what follows,

The path of integration in the definition (1.10) is a line in the complex

By introducing a new parameter

where, as before,

In the case when (for example)

which obviously yields the function

In analogy with the Definition (1.7), and motivated by their Definition (1.13) of the

and

which, in the special case when

The aim of this paper is to investigate the various properties of a

in two variables and the extended Lauricella’s hypergeometric function

of

Let a function

for some suitable constants

In terms of the function

and

provided that the defining integrals in Definitions (2.2-2.4) exist.

the definitions (2.2–2.4) immediately yield the definitions in (1.5–1.7) for the extended Gamma function

In terms of the function

For suitably constrained (real or complex) parameters

in two variables, and the extended Lauricella’s hypergeometric function:

of

and

where the generalized extended Beta function

and

where the generalized extended Beta function

We now proceed to derive integral representations for the above-defined hypergeometric functions in two and more variables.

as their Taylor–Maclaurin series, if we invert the order of summation and integration (which can easily be justified by absolute and uniform convergence), we find that

which, in view of the Definitions (1.6) and (2.7), yields the first member of the Assertion (2.13). Our demonstration of the integral Representation (2.13) is completed by applying the principle of analytic continuation, since the integral for

The proof of the Assertion (2.14) runs parallel to that of (2.13) and is based similarly upon the definitions (2.3) and (2.10) instead. The details involved are being omitted.

Theorems 2 and 3 below follow easily from the Definitions (1.6) and (2.3) in conjunction with the Definitions (2.8) and (2.11) and the Definitions (2.9) and (2.12), respectively.

it is easily seen that

which is rather instrumental in our demonstration of Theorem 2 along the lines of the proof of Theorem 1.

Earlier investigations by various authors dealing with operators of fractional calculus and their applications are adequately presented in the recent monograph [

and

where, as also in (1.13),

Making use of the Definition (3.2), we can easily derive the following analogue of the familiar fractional derivative Formula (1.11):

which would readily yield Theorem 4 below.

An interesting particular case of the fractional derivative Formula (3.5) asserted by Theorem 4 would occur when we specialize the sequence

We thus obtain the following interesting generalization of a known result [

provided that each member of

Since

this last result (3.7) can be written, in terms of the generalized extended Lauricella function

which, for

immediately yields the aforementioned known result [

Yet another result would emerge when, in the

so that, by using the definition (2.4), we have

Now, just as in our demonstration of the Assertion (3.5) of Theorem 4, if we apply the fractional derivative formula (3.3) (with

where we have also used the Definition (2.11) for the generalized extended Appell function

For

this last Formula (3.10) immediately yields a known result [

can be continued analytically for

Thus, clearly, in their special cases when

such

The Mellin transform of a suitably integrable function

whenever the improper integral in (4.1) exists.

_{2}_{1}

where we have also set

where we obviously have set

in the inner

Alternatively, by substituting from (3.3) into the left-hand side of (4.2), we have

which would lead us once again to the Assertion (4.2) of Theorem 5.

In order to prove the Mellin transform Formula (4.3), we first write

where we have used the already proven Assertion (4.2) of Theorem 5. The Assertion (4.3) of Theorem 5 would now follow upon interpreting the

Except for the obvious fact that the single

The Mellin transform Formula (4.3) corresponds to the case

In terms of the Lauricella hypergeometric function

which provides a multivariable hypergeometric extension of the Assertion (4.3) of Theorem 5. In particular, upon setting

in (4.2), if we make use of the Definition (3.1) (

which provides the

In this section, we derive linear and bilinear generating relations for the generalized extended hypergeometric functions in one, two and more variables (see

in the following form:

Now, upon multiplying both sides of (5.6) by

Interchanging the order of fractional differentiation and summation in (5.7), which can be justified when

we find from (5.7) that

which, by means of some obvious special cases of (3.8), yields the first Assertion (5.1) of Theorem 6 under the constraint derivable by appealing finally to the principle of analytic continuation.

Since

a

where we have only used the Definition (2.4) in conjunction with the expansion Formula (5.9).

The proof of the second Assertion (5.2) makes similar use of the generalized extended fractional derivative operator

instead of the Identity (5.5).

Next, upon setting

which, in light of (3.10) as well as some obvious special cases of (3.8), leads us eventually to the bilinear generating Relation (5.3) asserted by Theorem 6.

Finally, the proof of the Assertion (5.4) is much akin to that of (5.1). In fact, the role played by the argument

In our present investigation, we have introduced and studied a

It may be of interest to observe in conclusion that many of the definitions, which we have considered in this paper, can be

The corresponding

and

respectively. Moreover, the fractional derivative operator

where

Since

the definitions in (6.1) to (6.6) would obviously coincide with the corresponding definitions in the preceding sections when we set the