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.

Let a function be analytic within the disk and let its Taylor–Maclaurin coefficients be explicitly denoted (for convenience) by the sequence. Suppose also that the function can be continued analytically in the right half-plane with the asymptotic property given as follows:

for some suitable constants and depending essentially upon the sequence. Here, and in what follows, we assume that the series in the first part of the Definition (2.1) converges absolutely when for some and represents the function which is assumed to be analytic within the disk and which can be appropriately continued analytically elsewhere in the complex -plane with the order estimate provided in the second part of the Definition (2.1). For example, if we choose the sequence to be a suitable quotient of several -products with arguments linear in so that the function becomes identifiable with the familiar Fox–Wright -function, we can easily determine the radius of the above-mentioned disk and, moreover, we can then appropriately continue the resulting function analytically by means of a suitable Mellin–Barnes contour integral (see, for details [12, p. 56 et seq.]). Such functions as can indeed be specified on an ad hoc basis.

In terms of the function defined by (2.1), we now introduce a natural further generalization of the extended Gamma function , the extended Beta function and the extended hypergeometric function by

and

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

Remark 1. For various special choices of the sequence, the definitions in (2.2–2.4) would reduce to (known or new) extensions of the Gamma, Beta and hypergeometric functions. In particular, if we set

the definitions (2.2–2.4) immediately yield the definitions in (1.5–1.7) for the extended Gamma function , the extended Beta function and the extended hypergeometric function , respectively.

In terms of the function defined by definition (2.1), it is not difficult to generalize the integral representation definition (1.8) to the following form:

For suitably constrained (real or complex) parameters and , we propose these further generalizations of the extended Appell’s hypergeometric functions:

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

of variables , which are defined by

and

where the generalized extended Beta function is given by Definition (1.6). Clearly, the Definition (2.7) corresponds to the special case of the Definition (2.9) when . Moreover, in view of the relationship Definition (1.9), the Definitions (2.7) and (2.8) immediately yield the definitions in (1.14) and (1.15) when . More generally, in terms of the sequence defined involved in (2.1), we have the following definitions:

and

where the generalized extended Beta function is given by Definition (2.3).

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

Theorem 1. For the generalized extended Appell functions

defined by and the following integral representations hold true:

and

Proof. For convenience, we denote the second member of the Assertion (2.13) by and assume that . Then, upon expressing

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 above in (2.13) exists under the constraints which are listed already with (2.13).

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.

Theorem 2. For the functions

defined by and respectively, the following integral representations hold true:

and

Proof. Since [14, p. 52, Equation 1.6(2)]

it is easily seen that

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

Theorem 3. For the functions

defined by and respectively, the following integral representations hold true:

and

Proof. The proof of Theorem 3 is much akin to that of its special (two-variable) case (that is, Theorem 1) when . We, therefore, omit the details involved.

Earlier investigations by various authors dealing with operators of fractional calculus and their applications are adequately presented in the recent monograph [12] (see also [15]). The use of fractional derivative in the theory of generating functions is explained reasonably satisfactorily by Srivastava and Manocha (see, for details, [14, Chapter 5]). Here, in this section, we first introduce the following generalizations of the extended Riemann–Liouville fractional derivative operator defined by (1.13):

and

where, as also in (1.13), and the path of integration in each of the Definitions (3.1) and (3.2) is a line in the complex -plane from to .

Remark 2. The Definition (3.1) is easily recovered from (3.2) by specializing the sequence as in (2.5). Moreover, by using the specialization indicated in (1.9), the Definition (3.1) reduces immediately to (1.13). For , the Definitions (1.13), (3.1) and (3.2) would obviously reduce at once to the familiar Riemann–Liouville Definition (1.10). Each of these and the aforementioned other specializations are fairly straightforward. Henceforth, therefore, we choose to state our results in their general forms only and leave the specializations as an exercise for the interested reader.

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.

Theorem 4. In terms of a suitably bounded multiple sequence let the multivariable function be defined by

Then

provided that each member of exists.

Proof. The Assertion (3.5) of Theorem 4 follows easily from the Definitions (3.2) and (2.3). We, therefore, skip the details involved.

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

We thus obtain the following interesting generalization of a known result [14, p. 303, Problem 1]:

provided that each member of exists.

Since in the Definition (2.1), in its further special case when

this last result (3.7) can be written, in terms of the generalized extended Lauricella function defined by (2.12), as follows:

which, for or (alternatively) for

immediately yields the aforementioned known result [14, p. 303, Problem 1].

Yet another result would emerge when, in the two-variable () case of the Definition (3.4), we set

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 ) to times the -function given by (3.9), we are led to the following result:

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

For or (alternatively) for

this last Formula (3.10) immediately yields a known result [14, p. 289, Equation 5.1(18)].

Remark 3. The Beta function defined (for ) by

can be continued analytically for as follows (see, for example, [14, p. 26, Equation 1.1(48)]):

Thus, clearly, in their special cases when

such additional constraints as in (3.3), (3.5) and (3.7), and in (3.8) and (3.10), can be dropped fairly easily by applying both cases of the definition in (3.2).

The Mellin transform of a suitably integrable function with index is defined, as usual, by

whenever the improper integral in (4.1) exists.

Theorem 5. In terms of the generalized extended Gamma function defined by the Mellin transforms of the following generalized extended fractional derivatives defined by are given by

and

And , more generally , by

provided that each member of the Assertions (4.2), (4.3) and (4.4) exists, ₂F₁ being the Gauss hypergeometric function.

Proof. Using the Definition (4.1) of the Mellin transform, we find from (3.2) that

where we have also set and in the inner -integral. Upon interchanging the order of integration in (4.5), which can easily be justified by absolute convergence of the integrals involved under the constraints state with (4.2), we get

where we obviously have set

in the inner -integral. We now interpret the -integral and the -integral in (4.6) by means of the Definitions (2.2) (with ) and (3.12), respectively. This evidently completes our derivation of the Mellin transform Formula (4.2) asserted by Theorem 5.

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 -series in the last member of (4.7) as a Gauss hypergeometric function .

Except for the obvious fact that the single -series is to be replaced by the multiple -series, the demonstration of the third Assertion (4.4) of Theorem 5 would run parallel to that of the second Assertion (4.3). The details involved may thus be omitted here.

The Mellin transform Formula (4.3) corresponds to the case of the general Result (4.4). Moreover, in its special case when (or when ), (4.3) would reduce at once to the Mellin transform Formula (4.2).

In terms of the Lauricella hypergeometric function of variables (see, for details, [14, p. 60, Equation 1.7(4)], the special case of the assertion (4.4) of Theorem 5 when yields the following Mellin transform formula:

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) (with ), we obtain

which provides the duly-corrected version of a known result asserted recently by Özarslan and Özergin [1, p. 1832, Theorem 4.2].

In this section, we derive linear and bilinear generating relations for the generalized extended hypergeometric functions in one, two and more variables (see Section 2) by following the methods which are described fairly adequately in the monograph by Srivastava and Manocha [14, Chapter 5]. Our main results are contained in Theorem 6 below.

Theorem 6. Each of the following generating relations holds true for the generalized extended hypergeometric functions in one and more variables:

and

provided that each member of the generating relations to exists.

Proof. Our demonstration of Theorem 6 is based upon the generalized extended fractional derivative operator defined by (3.2). We first rewrite the elementary identity:

in the following form:

Now, upon multiplying both sides of (5.6) by , if we apply the generalized extended fractional derivative operator on each member of the resulting equation, we find that

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 direct proof of the generating relation (5.1), without using the generalized extended fractional derivative operator defined by (3.2), can be given along the following lines:

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 defined by (3.2) together with the following elementary identity:

instead of the Identity (5.5).

Next, upon setting and in (5.1), if we multiply the resulting equation by and then apply the generalized extended fractional derivative operator together with the elementary Identity (5.11), we find that

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 (5.4) can be assumed instead by any of the other arguments .

In our present investigation, we have introduced and studied a further generalization of the extended fractional derivative operator related to a generalized Beta function, which was used in order to obtain some linear and bilinear generating relations involving the extended hypergeometric functions [1]. We have applied the generalized extended fractional derivative operator to derive generating relations for the generalized extended Gauss, Appell and Lauricella hypergeometric functions in one, two and more variables. Many other properties and relationships involving (for example) Mellin transforms and the generalized extended fractional derivative operator are also given.

It may be of interest to observe in conclusion that many of the definitions, which we have considered in this paper, can be further extended by introducing one additional parameter (with . Thus, in terms of the -function given by (2.1), we can introduce a further extension of the generalized extended Beta function in (2.3) as follows:

The corresponding further extensions of the Definitions (2.4) and (2.10) to (2.12) are given by

and

respectively. Moreover, the fractional derivative operator defined by (3.2) can be further extended as follows:

where and, as also in (1.10), (1.13), (3.1) and (3.2), the path of integration in the Definition (6.6) is a line in the complex -plane from to .

Since

the definitions in (6.1) to (6.6) would obviously coincide with the corresponding definitions in the preceding sections when we set the additional parameter . Most (if not all) of the properties and results, which we have investigated in this paper in the case, can indeed be considered analogously for the case in a rather simple and straightforward manner. The details involved may, therefore, be left as an exercise for the interested reader.