2. The Generalized Extended Appell and Lauricella Functions
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.
3. Applications of the Generalized Extended Riemann–Liouville Fractional Derivative Operator
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).
4. Mellin Transforms of the Generalized Extended Fractional Derivatives
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].
5. A Set of Generating Functions
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
.
6. Concluding Remarks and Observations
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.