Axioms 2012, 1(3), 238-258; doi:10.3390/axioms1030238

Article
A Class of Extended Fractional Derivative Operators and Associated Generating Relations Involving Hypergeometric Functions
H. M. Srivastava 1,*, Rakesh K. Parmar 2 and Purnima Chopra 3
1
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
2
Department of Mathematics, Government College of Engineering and Technology, Bikaner 334004, Rajasthan, India; Email: rakeshparmar27@gmail.com
3
Department of Mathematics, Marudhar Engineering College, Raisar, NH-11 Jaipur Road, Bikaner 334001, Rajasthan, India; Email: purnimachopra@rediffmail.com
*
Author to whom correspondence should be addressed; Email: harimsri@math.uvic.ca; Tel.: +1-250-477-6960 or +1-250-472-5313; Fax: +1-250-721-8962.
Received: 29 June 2012; in revised form: 8 September 2012 / Accepted: 12 September 2012 /
Published: 5 October 2012

Abstract

: 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 [1]. The main object of this paper is to present a further generalization of the extended fractional derivative operator and apply the generalized extended fractional derivative operator to derive linear and bilinear generating relations for the generalized extended Gauss, Appell and Lauricella hypergeometric functions in one, two and more variables. Some other properties and relationships involving the Mellin transforms and the generalized extended fractional derivative operator are also given.
Keywords:
gamma and beta functions; Eulerian integrals; generating functions; hypergeometric functions; Appell-Lauricella hypergeometric functions; fractional derivative operators; Mellin transforms

1. Introduction, Definitions and Preliminaries

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.

1.1. The Extended Beta, Gamma and Hypergeometric Functions

Extensions of a number of well-known special functions were investigated recently by several authors (see, for example [2,3,4,5,6,7]). In particular, Chaudhry et al. [3] gave the following interesting extension of the classical Beta function Axioms 01 00238 i002:

Axioms 01 00238 i003

so that, clearly,

Axioms 01 00238 i005

Here, and in what follows, such arguments as (for example) Axioms 01 00238 i006 in the Definition (1.1) are motivated by the connection of the extended Beta function Axioms 01 00238 i007 with the Macdonald (or modified Bessel) function Axioms 01 00238 i008 (see, for details, [8,9]).

Making use of the extended Beta function Axioms 01 00238 i007 defined by (1.1), Chaudhry et al. [8] introduced the extended hypergeometric function as follows:

Axioms 01 00238 i009

where Axioms 01 00238 i011 denotes the Pochhammer symbol or the shifted factorial, which is defined (for Axioms 01 00238 i012 and Axioms 01 00238 i013 by

Axioms 01 00238 i014

it being understood conventionally that Axioms 01 00238 i015.

Among several interesting and potentially useful properties of the extended hypergeoemetric function Axioms 01 00238 i016 defined by (1.2), the following integral representation of the Pfaff–Kummer type was also given by Chaudhry et al. [8, p. 592, Equation (3.2)]:

Axioms 01 00238 i017
Axioms 01 00238 i018

Obviously, for the Gauss hypergeometric function Axioms 01 00238 i019, we have

Axioms 01 00238 i020

The following further generalizations of the extended Gamma function Axioms 01 00238 i021 (see, for details [9]), the extended Beta function Axioms 01 00238 i022 and the extended Gauss hypergeometric function Axioms 01 00238 i023 were considered more recently by Özergin et al. [7]:

Axioms 01 00238 i024
Axioms 01 00238 i026

and

Axioms 01 00238 i028

respectively, Axioms 01 00238 i030 being the (Kummer’s) confluent hypergeometric function. The following integral representation of the Pfaff–Kummer type was also given by Özergin et al. [7]:

Axioms 01 00238 i031

Clearly, since

Axioms 01 00238 i033

the special cases of Equations (1.6–1.8) when Axioms 01 00238 i034 would immediately yield Equations (1.1), (1.2) and (1.4), respectively.

Throughout our present investigation, it is tacitly assumed that Axioms 01 00238 i035 and various other lower (or denominator) parameters are not zero or negative integers (that is, no zeros appear in the denominators).

1.2. The Riemann–Liouville Fractional Derivative Operator and Its Generalizations

For the Riemann–Liouville fractional derivative operator Axioms 01 00238 i036 defined by (see, for example [10, p. 181] [11] and [12, p. 70 et seq.])

Axioms 01 00238 i037

it is known that

Axioms 01 00238 i038

where, and in what follows,

Axioms 01 00238 i039

The path of integration in the definition (1.10) is a line in the complex Axioms 01 00238 i040-plane from Axioms 01 00238 i041 to Axioms 01 00238 i042.

By introducing a new parameter Axioms 01 00238 i043 of the type which is involved in (for example) the Definitions (1.1) and (1.2), Özarslan and Özergin [1] defined the correspondingly extended Riemann–Liouville fractional derivative operator Axioms 01 00238 i044 by

Axioms 01 00238 i045

where, as before, Axioms 01 00238 i046. The path of integration in the Definition (1.13), which immediately yields the definition (1.10) when Axioms 01 00238 i047, is also a line in the complex Axioms 01 00238 i040-plane from Axioms 01 00238 i041 to Axioms 01 00238 i042. The argument Axioms 01 00238 i048 in the Definition (1.13) and elsewhere in this paper is obviously necessitated by the applicability of the definition (1.1) of the extended Beta function Axioms 01 00238 i007 when we set

Axioms 01 00238 i049

In the case when (for example) Axioms 01 00238 i050, we find from the second part of the Definition (1.13) (with Axioms 01 00238 i051) that

Axioms 01 00238 i052

which obviously yields the function Axioms 01 00238 i053 when Axioms 01 00238 i047. Thus, in general, the natural connection of the Riemann–Liouville fractional derivative operator Axioms 01 00238 i054 defined by (1.10) with ordinary derivatives when the order Axioms 01 00238 i055 is zero or a positive integer is lost by the extended fractional derivative operator in Definition (1.13) and its further generalizations which we have considered in our present investigation.

1.3. Extended Appell Hypergeometric Functions in Two Variables

In analogy with the Definition (1.7), and motivated by their Definition (1.13) of the extended Riemann–Liouville fractional derivative Axioms 01 00238 i044, Özarslan and Özergin [1] extended the familiar Appell hypergeometric functions Axioms 01 00238 i056 and Axioms 01 00238 i057 in two variables as follows:

Axioms 01 00238 i058

and

Axioms 01 00238 i060

which, in the special case when Axioms 01 00238 i047, yield the familiar Appell hypergeometric functions Axioms 01 00238 i056 and Axioms 01 00238 i057 in two variables (see [13, p. 14]). For each of these extended Appell hypergeometric functions, such properties as their integral representations and relationships with the extended Riemann–Liouville fractional derivative operator Axioms 01 00238 i062 defined by (1.13) can also be found in the work of Özarslan and Özergin [1].

The aim of this paper is to investigate the various properties of a further generalization of the extended fractional derivative operator Axioms 01 00238 i062 defined by (1.13) and apply the generalized operator to derive generating relations for hypergeometric functions in one, two and more variables. We first introduce, in Section 2, the following further generalizations of the extended Appell’s hypergeometric functions:

Axioms 01 00238 i063

in two variables and the extended Lauricella’s hypergeometric function

Axioms 01 00238 i064

of Axioms 01 00238 i065 variables Axioms 01 00238 i066 are defined and their integral representations are obtained. In Section 3, we introduce and study the properties and relationships associated with the above-mentioned further generalization of the extended fractional derivative operator Axioms 01 00238 i062 defined by Definition (1.13) and apply the generalized operator in order to obtain various generating relations in terms of the generalized extended Appell and Lauricella hypergeometric functions in two and more variables. Section 4 contains some results related to the Mellin transforms and the extended fractional derivative operator. In Section 5, some generating relations for generalized extended hypergeometric functions are obtained via the above-mentioned further generalized fractional derivative operator by following the lines which are detailed in the monograph by Srivastava and Manocha [14]. Finally, in Section 6, we conclude this paper by presenting a number of remarks and observations pertaining to our investigation here.

2. The Generalized Extended Appell and Lauricella Functions

Let a function Axioms 01 00238 i067 be analytic within the disk Axioms 01 00238 i068 and let its Taylor–Maclaurin coefficients be explicitly denoted (for convenience) by the sequence Axioms 01 00238 i069. Suppose also that the function Axioms 01 00238 i070 can be continued analytically in the right half-plane Axioms 01 00238 i071 with the asymptotic property given as follows:

Axioms 01 00238 i072

for some suitable constants Axioms 01 00238 i073 and Axioms 01 00238 i074 depending essentially upon the sequence Axioms 01 00238 i075. Here, and in what follows, we assume that the series in the first part of the Definition (2.1) converges absolutely when Axioms 01 00238 i076 for some Axioms 01 00238 i077 and represents the function Axioms 01 00238 i078 which is assumed to be analytic within the disk Axioms 01 00238 i068 and which can be appropriately continued analytically elsewhere in the complex Axioms 01 00238 i079-plane with the order estimate provided in the second part of the Definition (2.1). For example, if we choose the sequence Axioms 01 00238 i080 to be a suitable quotient of several Axioms 01 00238 i081-products with arguments linear in Axioms 01 00238 i082 so that the function Axioms 01 00238 i083 becomes identifiable with the familiar Fox–Wright Axioms 01 00238 i084-function, we can easily determine the radius Axioms 01 00238 i085 of the above-mentioned disk and, moreover, we can then appropriately continue the resulting function Axioms 01 00238 i086 analytically by means of a suitable Mellin–Barnes contour integral (see, for details [12, p. 56 et seq.]). Such functions as Axioms 01 00238 i087 can indeed be specified on an ad hoc basis.

In terms of the function Axioms 01 00238 i088 defined by (2.1), we now introduce a natural further generalization of the extended Gamma function Axioms 01 00238 i089, the extended Beta function Axioms 01 00238 i090 and the extended hypergeometric function Axioms 01 00238 i091 by

Axioms 01 00238 i092
Axioms 01 00238 i094

and

Axioms 01 00238 i096

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

Remark 1. For various special choices of the sequence Axioms 01 00238 i098, 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

Axioms 01 00238 i099

the definitions (2.2–2.4) immediately yield the definitions in (1.5–1.7) for the extended Gamma function Axioms 01 00238 i089, the extended Beta function Axioms 01 00238 i090 and the extended hypergeometric function Axioms 01 00238 i091, respectively.

In terms of the function Axioms 01 00238 i100 defined by definition (2.1), it is not difficult to generalize the integral representation definition (1.8) to the following form:

Axioms 01 00238 i101
Axioms 01 00238 i279

For suitably constrained (real or complex) parameters Axioms 01 00238 i102 and Axioms 01 00238 i035, we propose these further generalizations of the extended Appell’s hypergeometric functions:

Axioms 01 00238 i280

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

Axioms 01 00238 i281

of Axioms 01 00238 i065 variables Axioms 01 00238 i066, which are defined by

Axioms 01 00238 i103
Axioms 01 00238 i104
Axioms 01 00238 i105
Axioms 01 00238 i106

and

Axioms 01 00238 i107
Axioms 01 00238 i108

where the generalized extended Beta function Axioms 01 00238 i090 is given by Definition (1.6). Clearly, the Definition (2.7) corresponds to the special case of the Definition (2.9) when Axioms 01 00238 i109. 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 Axioms 01 00238 i110. More generally, in terms of the sequence Axioms 01 00238 i098 defined involved in (2.1), we have the following definitions:

Axioms 01 00238 i111
Axioms 01 00238 i282
Axioms 01 00238 i112
Axioms 01 00238 i283

and

Axioms 01 00238 i113
Axioms 01 00238 i114

where the generalized extended Beta function Axioms 01 00238 i115 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

Axioms 01 00238 i116

defined by Axioms 01 00238 i117 and Axioms 01 00238 i118 the following integral representations hold true:

Axioms 01 00238 i120
Axioms 01 00238 i121
Axioms 01 00238 i122

and

Axioms 01 00238 i123
Axioms 01 00238 i124

Proof. For convenience, we denote the second member of the Assertion (2.13) by Axioms 01 00238 i125 and assume that Axioms 01 00238 i126. Then, upon expressing

Axioms 01 00238 i127

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

Axioms 01 00238 i128
Axioms 01 00238 i129
Axioms 01 00238 i130

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 Axioms 01 00238 i125 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

Axioms 01 00238 i131

defined by Axioms 01 00238 i132 and Axioms 01 00238 i133 respectively, the following integral representations hold true:

Axioms 01 00238 i135
Axioms 01 00238 i136
Axioms 01 00238 i137

and

Axioms 01 00238 i138
Axioms 01 00238 i139
Axioms 01 00238 i140

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

Axioms 01 00238 i141

it is easily seen that

Axioms 01 00238 i142
Axioms 01 00238 i143

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

Theorem 3. For the functions

Axioms 01 00238 i144

defined by Axioms 01 00238 i145 and Axioms 01 00238 i146 respectively, the following integral representations hold true:

Axioms 01 00238 i147
Axioms 01 00238 i284
Axioms 01 00238 i285

and

Axioms 01 00238 i149
Axioms 01 00238 i151

Proof. The proof of Theorem 3 is much akin to that of its special (two-variable) case (that is, Theorem 1) when Axioms 01 00238 i109. 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 Axioms 01 00238 i152 defined by (1.13):

Axioms 01 00238 i153

and

Axioms 01 00238 i154

where, as also in (1.13), Axioms 01 00238 i155 and the path of integration in each of the Definitions (3.1) and (3.2) is a line in the complex Axioms 01 00238 i040-plane from Axioms 01 00238 i041 to Axioms 01 00238 i042.

Remark 2. The Definition (3.1) is easily recovered from (3.2) by specializing the sequence Axioms 01 00238 i075 as in (2.5). Moreover, by using the specialization indicated in (1.9), the Definition (3.1) reduces immediately to (1.13). For Axioms 01 00238 i047, 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):

Axioms 01 00238 i156

which would readily yield Theorem 4 below.

Theorem 4. In terms of a suitably bounded multiple sequence Axioms 01 00238 i157 let the multivariable function Axioms 01 00238 i158 be defined by

Axioms 01 00238 i159

Then

Axioms 01 00238 i160
Axioms 01 00238 i161
Axioms 01 00238 i162

provided that each member of Axioms 01 00238 i163 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 Axioms 01 00238 i164 as follows:

Axioms 01 00238 i165

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

Axioms 01 00238 i166
Axioms 01 00238 i167

provided that each member of Axioms 01 00238 i168 exists.

Since Axioms 01 00238 i169 in the Definition (2.1), in its further special case when

Axioms 01 00238 i170

this last result (3.7) can be written, in terms of the generalized extended Lauricella function Axioms 01 00238 i171 defined by (2.12), as follows:

Axioms 01 00238 i172
Axioms 01 00238 i173

which, for Axioms 01 00238 i047 or (alternatively) for

Axioms 01 00238 i174

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

Yet another result would emerge when, in the two-variable ( Axioms 01 00238 i109) case of the Definition (3.4), we set

Axioms 01 00238 i175

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

Axioms 01 00238 i176

Now, just as in our demonstration of the Assertion (3.5) of Theorem 4, if we apply the fractional derivative formula (3.3) (with Axioms 01 00238 i177) to Axioms 01 00238 i178 times the Axioms 01 00238 i179-function given by (3.9), we are led to the following result:

Axioms 01 00238 i180
Axioms 01 00238 i181

where we have also used the Definition (2.11) for the generalized extended Appell function Axioms 01 00238 i182.

For Axioms 01 00238 i047 or (alternatively) for

Axioms 01 00238 i183

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

Remark 3. The Beta function Axioms 01 00238 i002 defined (for Axioms 01 00238 i184) by

Axioms 01 00238 i185

can be continued analytically for Axioms 01 00238 i186 as follows (see, for example, [14, p. 26, Equation 1.1(48)]):

Axioms 01 00238 i187
Axioms 01 00238 i188

Thus, clearly, in their special cases when

Axioms 01 00238 i189

such additional constraints as Axioms 01 00238 i190 in (3.3), (3.5) and (3.7), and Axioms 01 00238 i191 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 Axioms 01 00238 i192 with index Axioms 01 00238 i193 is defined, as usual, by

Axioms 01 00238 i194

whenever the improper integral in (4.1) exists.

Theorem 5. In terms of the generalized extended Gamma function Axioms 01 00238 i195 defined by Axioms 01 00238 i196 the Mellin transforms of the following generalized extended fractional derivatives defined by Axioms 01 00238 i197 are given by

Axioms 01 00238 i198
Axioms 01 00238 i199

and

Axioms 01 00238 i200
Axioms 01 00238 i201
Axioms 01 00238 i202

And, more generally, by

Axioms 01 00238 i203
Axioms 01 00238 i204
Axioms 01 00238 i205

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

Axioms 01 00238 i209

where we have also set Axioms 01 00238 i210 and Axioms 01 00238 i211 in the inner Axioms 01 00238 i040-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

Axioms 01 00238 i212

where we obviously have set

Axioms 01 00238 i213

in the inner Axioms 01 00238 i043-integral. We now interpret the Axioms 01 00238 i214-integral and the Axioms 01 00238 i215-integral in (4.6) by means of the Definitions (2.2) (with Axioms 01 00238 i047) 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

Axioms 01 00238 i216
Axioms 01 00238 i286

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

Axioms 01 00238 i217
Axioms 01 00238 i287

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 Axioms 01 00238 i218-series in the last member of (4.7) as a Gauss hypergeometric function Axioms 01 00238 i019.

Except for the obvious fact that the single Axioms 01 00238 i218-series is to be replaced by the multiple Axioms 01 00238 i219-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 Axioms 01 00238 i220 of the general Result (4.4). Moreover, in its special case when Axioms 01 00238 i221 (or when Axioms 01 00238 i222), (4.3) would reduce at once to the Mellin transform Formula (4.2).

In terms of the Lauricella hypergeometric function Axioms 01 00238 i223 of Axioms 01 00238 i065 variables (see, for details, [14, p. 60, Equation 1.7(4)], the special case of the assertion (4.4) of Theorem 5 when Axioms 01 00238 i224 yields the following Mellin transform formula:

Axioms 01 00238 i225
Axioms 01 00238 i226
Axioms 01 00238 i227

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

Axioms 01 00238 i228

in (4.2), if we make use of the Definition (3.1) (with Axioms 01 00238 i110), we obtain

Axioms 01 00238 i229
Axioms 01 00238 i230
Axioms 01 00238 i231

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:

Axioms 01 00238 i232
Axioms 01 00238 i233
Axioms 01 00238 i234

and

Axioms 01 00238 i235

provided that each member of the generating relations Axioms 01 00238 i236 to Axioms 01 00238 i237 exists.

Proof. Our demonstration of Theorem 6 is based upon the generalized extended fractional derivative operator Axioms 01 00238 i238 defined by (3.2). We first rewrite the elementary identity:

Axioms 01 00238 i239

in the following form:

Axioms 01 00238 i240

Now, upon multiplying both sides of (5.6) by Axioms 01 00238 i241, if we apply the generalized extended fractional derivative operator Axioms 01 00238 i242 on each member of the resulting equation, we find that

Axioms 01 00238 i243

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

Axioms 01 00238 i244

we find from (5.7) that

Axioms 01 00238 i245

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

Axioms 01 00238 i246

a direct proof of the generating relation (5.1), without using the generalized extended fractional derivative operator Axioms 01 00238 i247 defined by (3.2), can be given along the following lines:

Axioms 01 00238 i248

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 Axioms 01 00238 i249 defined by (3.2) together with the following elementary identity:

Axioms 01 00238 i250

instead of the Identity (5.5).

Next, upon setting Axioms 01 00238 i251 and Axioms 01 00238 i252 in (5.1), if we multiply the resulting equation by Axioms 01 00238 i253 and then apply the generalized extended fractional derivative operator Axioms 01 00238 i254 together with the elementary Identity (5.11), we find that

Axioms 01 00238 i255

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 Axioms 01 00238 i256 in (5.4) can be assumed instead by any of the other arguments Axioms 01 00238 i257.

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 Axioms 01 00238 i258 (with Axioms 01 00238 i259. Thus, in terms of the Axioms 01 00238 i260-function given by (2.1), we can introduce a further extension of the generalized extended Beta function in (2.3) as follows:

Axioms 01 00238 i261
Axioms 01 00238 i262

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

Axioms 01 00238 i263
Axioms 01 00238 i264
Axioms 01 00238 i265
Axioms 01 00238 i266
Axioms 01 00238 i267
Axioms 01 00238 i268

and

Axioms 01 00238 i269
Axioms 01 00238 i270

respectively. Moreover, the fractional derivative operator Axioms 01 00238 i271 defined by (3.2) can be further extended as follows:

Axioms 01 00238 i272

where Axioms 01 00238 i273 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 Axioms 01 00238 i040-plane from Axioms 01 00238 i041 to Axioms 01 00238 i042.

Since

Axioms 01 00238 i274

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 Axioms 01 00238 i275. Most (if not all) of the properties and results, which we have investigated in this paper in the Axioms 01 00238 i276 case, can indeed be considered analogously for the Axioms 01 00238 i277 case in a rather simple and straightforward manner. The details involved may, therefore, be left as an exercise for the interested reader.

References

  1. Özarslan, M.A.; Özergin, E. Some generating relations for extended hypergeometric function via generalized fractional derivative operator. Math. Comput. Model. 2010, 52, 1825–1833, doi:10.1016/j.mcm.2010.07.011.
  2. Chaudhry, M.A.; Temme, N.M.; Veling, E.J.M. Asymptotic and closed form of a generalized incomplete gamma function. J. Comput. Appl. Math. 1996, 67, 371–379, doi:10.1016/0377-0427(95)00018-6.
  3. Chaudhry, M.A.; Qadir, A.; Rafique, M.; Zubair, S.M. Extension of Euler’s beta function. J. Comput. Appl. Math. 1997, 78, 19–32, doi:10.1016/S0377-0427(96)00102-1.
  4. Chaudhry, M.A.; Zubair, S.M. Generalized incomplete gamma functions with applications. J. Comput. Appl. Math. 1994, 55, 99–124, doi:10.1016/0377-0427(94)90187-2.
  5. Chaudhry, M.A.; Zubair, S.M. On the decomposition of generalized incomplete gamma functions with applications of Fourier transforms. J. Comput. Appl. Math. 1995, 59, 253–284, doi:10.1016/0377-0427(94)00026-W.
  6. Miller, A.R. Reduction of a generalized incomplete gamma function, related Kampé de Fériet functions, and incomplete Weber integrals. Rocky Mountain J. Math. 2000, 30, 703–714, doi:10.1216/rmjm/1022009290.
  7. Özergin, E.; Özarslan, M.A.; Altn, A. Extension of gamma, beta and hypergeometric functions. J. Comput. Appl. Math. 2011, 235, 4601–4610, doi:10.1016/j.cam.2010.04.019.
  8. Chaudhry, M.A.; Qadir, A.; Srivastava, H.M.; Paris, R.B. Extended hypergeometric and confluent hypergeometric functions. Appl. Math. Comput. 2004, 159, 589–602, doi:10.1016/j.amc.2003.09.017.
  9. Chaudhry, M.A.; Zubair, S.M. On a Class of Incomplete Gamma Functions with Applications; CRC Press (Chapman and Hall): Boca Raton, FL, USA, 2002.
  10. Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G. Tables of Integral Transforms, Volume II; McGraw-Hill Book Company: New York, NY, USA, 1954.
  11. Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications, Translated from the Russian: Integrals and Derivatives of Fractional Order and Some of Their Applications (“Nauka i Tekhnika", Minsk, 1987); Gordon and Breach Science Publishers: Reading, UK, 1993.
  12. Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations. North-Holland Mathematical Studies; Elsevier (North-Holland) Science Publishers: Amsterdam, The Netherlands, 2006; Volume 204.
  13. Appell, P.; Kampé de Fériet, J. Fonctions Hypergéométriques et Hypersphériques: Polynômes d’Hermite; Gauthier-Villars: Paris, France, 1926.
  14. Srivastava, H.M.; Manocha, H.L. A Treatise on Generating Functions; Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons: New York, NY, USA, 1984.
  15. Srivastava, H.M.; Saxena, R.K. Operators of fractional integration and their applications. Appl. Math. Comput. 2001, 118, 1–52, doi:10.1016/S0096-3003(99)00208-8.
Axioms EISSN 2075-1680 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert