Abstract
The main purpose of this paper is to introduce a class of new extended forms of the beta function, Gauss hypergeometric function and Appell-Lauricella hypergeometric functions by means of the modified Bessel function of the third kind. Some typical generating relations for these extended hypergeometric functions are obtained by defining the extension of the Riemann-Liouville fractional derivative operator. Their connections with elementary functions and Fox’s H-function are also presented.
1 Introduction
Extensions and generalizations of some known special functions are important both from the theoretical and applied point of view. Also many extensions of fractional derivative operators have been developed and applied by many authors (see [2, 3, 4, 5, 6, 11, 12, 19, 20, 21] and [17, 18]). These new extensions have proved to be very useful in various fields such as physics, engineering, statistics, actuarial sciences, economics, finance, survival analysis, life testing and telecommunications. The above-mentioned applications have largely motivated our present study.
The extended incomplete gamma functions constructed by using the exponential function are defined by
and
with |arg z| < π, which have been studied in detail by Chaudhry and Zubair (see, for example, [2] and [4]). The extended incomplete gamma functions γ(α, z; p) and Γ(α, z; p) satisfy the following decomposition formula
where Γp(α) is called extended gamma function, and Kα(z) is the modified Bessel function of the third kind, or the Macdonald function with its integral representation given by (see [7])
where ℜ(z) > 0 and
For α =
Instead of using the exponential function, Chaudhry and Zubair extended (1) and (2) in the following form (see [3], see also [4])
and
where ℜ(x) > 0, ℜ(p) > 0, −∞ < α < ∞.
Inspired by their construction of (7) and (8), we aim to introduce a class of new special functions and fractional derivative operator by suitably using the modified Bessel function Kα(z).
The present paper is organized as follows: In Section 2, we first define the extended beta function and study some of its properties such as different integral representations and its Mellin transform. Then some extended hypergeometric functions are introduced by using the extended beta function. The extended Riemann-Liouville type fractional derivative operator and its properties are given in Section 3. In Section 4, the linear and bilinear generating relations for the extended hypergeometric functions are derived. Finally, the Mellin transforms of the extended fractional derivative operator are determined in Section 5.
2 Extended beta and hypergeometric functions
This section is divided into two subsections. In subsection-1, we define the extended beta function Bμ(x, y; p; m) and study some of its properties. In subsection-2, we introduce the extended Gauss
hypergeometric function Fμ(a, b; c; z; p; m), the Appell hypergeometric functions F1,μ, F2,μ and the Lauricella hypergeometric function
2.1 Extended beta function
Definition 2.1
The extended beta function Bμ(x, y; p; m) with ℜ(p) > 0 is defined by
where x, y ∈ ℂ, m > 0 and ℜ(μ) ≥ 0.
Remark 2.2
Taking m = 1, μ = 0 and making use of (6), (9) reduces to the extended beta function Bμ(x, y; p) defined by Chaudhry et al. [5, Eq. (1.7)]
where ℜ(p) > 0 and x, y ∈ ℂ.
Theorem 2.3
The following integral representations for the extended beta functions Bμ(x, y; p; m) with ℜ(p) > 0 are valid
Proof
These formulas can be obtained by using the transformations
Theorem 2.4
The following expression holds true
where κ(p|u) is given by (5).
Proof
Expressing Bμ(x, y; p, m) in its integral form with the help of (9), and taking (4) into account, we obtain
where κ(p|u) is given by (5).
In order to write the inner integral as our extended beta function, we need the following variant of (6), that is,
Then, we have
Substituting (16) into (15) we obtain the required result (14).□
Remark 2.5
It is interesting to note that using the definition of the extended beta function (see, [11, 12]) we can get the following expression for (9)
where the function Bb;ρ,λ(x, y) is given by [12, p. 631, Eq. (2)]
The following theorem establishes the relation between the Mellin transform
and the extended beta function.
Theorem 2.6
Let x, y ∈ ℂ, m > 0, ℜ(μ) ≥ 0 and
Then we have the following relation
Proof
First, we have
Since the Mellin transform of the Macdonald function Kv(z) is given by [9, p. 37, Eq.(1.7.41)]:
the last integral in (19) can be evaluated as
where we have used
Finally, we get
□
Now we can derive the Fox H-function representation of the extended beta function defined in (9).
Let m, n, p, q be integers such that 0 ≥ m ≥ q, 0 ≥ n ≥ p, and for parameters ai, bi ∈ ℂ and for parameters αi, βj ∈ ℝ+ (i = 1, …, p; j = 1, …, q), the H-function is defined in terms of a Mellin-Barnes integral in the following manner ([8, pp. 1–2]; see also [10, p. 343, Definition E.1.] and [13, p. 2, Definition 1.1.]):
where
with the contour 𝓛 suitably chosen. As convention, the empty product is equal to one. The theory of the H-function is well explained in the books of Mathai [14], Mathai and Saxena ([15], Ch.2), Srivastava, Gupta and Goyal ([22], Ch.1) and Kilbas and Saigo ([8], Ch.1 and Ch.2). Note that (18) and (20) mean
Theorem 2.7
Let ℜ(p) > 0, x, y ∈ ℂ, m > 0 and ℜ(μ) ≥ 0, then
where Bμ(x, y; p; m) is as defined in (9).
Proof
The result is obtained by taking the inverse Mellin transform of (18) in Theorem 2.6 and using (22) and (23).□
2.2 Extended hypergeometric functions
Definition 2.8
The extended Gauss hypergeometric function Fμ(a, b; c; z; p; m) is defined by
where ℜ(p) > 0, ℜ(μ) ≥ 0, 0 < ℜ(b) < ℜ(c) and |z| < 1.
Definition 2.9
The extended Appell hypergeometric function F1,μ is defined by
where ℜ(p) > 0, ℜ(μ) ≥ 0, 0 < ℜ(a) < ℜ(d) and |x| < 1, |y| < 1.
Definition 2.10
The extended Appell hypergeometric function F2,μ is defined by
where ℜ(p) > 0, ℜ(μ) ≥ 0, 0 < ℜ(b) < ℜ(d), 0 < ℜ(c) < ℜ(e) and |x| + |y| < 1.
Definition 2.11
The extended Lauricella hypergeometric function
where ℜ(p) > 0, ℜ(μ) ≥ 0, 0 < ℜ(a) < ℜ(e) and |x| < 1, |y| < 1, |z| < 1.
Here, it is important to mention that when we take m = 1, μ = 0 and then letting p → 0, function (24) reduces to the ordinary Gauss hypergeometric function defined by
where (x)n denotes the Pochhammer symbol defined, in terms of the familiar gamma function, by
For conditions of convergence and other related details of this function, see [1], [9] and [16]. Similarly, we can reduce the functions (25), (26) and (27) to the well-known Appell functions F1, F2 and Lauricella function
Now, we establish the integral representations of the extended hypergeometric functions given by (24), (25), (26) and (27) as follows.
Theorem 2.12
The following integral representation for the extended Gauss hypergeometric function Fμ(a, b; c; z; p; m) is valid
where |arg (1 − z)| < π, ℜ(p) > 0, m > 0 and ℜ(μ) ≥ 0.
Proof
By using (9) and employing the binomial expansion
we get the above integral representation.□
Theorem 2.13
The following integral representation for the extended hypergeometric function F1,μ is valid
Proof
For simplicity, let ℑ denote the left-hand side of (31). Then, using (25) yields
By applying (9) to the integrand of (31), after a little simplification, we have
By interchanging the order of summation and integration in (33), we get
which proves the integral representation (31).□
To establish Theorem 2.13, we need to recall the following elementary series identity involving the bounded sequence of
Lemma 2.14
For a bounded sequence
Theorem 2.15
The following integral representation for the extended hypergeometric function F2,μ is valid
Proof
Let 𝓛 denote the left-hand side of (36). Then, using (26) yields
By applying (9) to the integrand of (32), we have
Next, interchanging the order of summation and integration in (38), which is guaranteed, yields
Finally, applying (35) to the double series in (39), we obtain the right-hand side of (36).□
Theorem 2.16
The following integral representation for the extended hypergeometric function
Proof
A similar argument in the proof of Theorem 2.15 will be able to establish the integral representation in (40). Therefore, details of the proof are omitted.□
3 Extended Riemann-Liouville fractional derivative operator
We first recall that the classical Riemann-Liouville fractional derivative is defined by (see [23, p. 286])
where ℜ(ν) < 0 and the integration path is a line from 0 to z in the complex t-plane. It coincides with the fractional integral of order −ν. In case m − 1 < ℜ(ν) < m, m ∈ ℕ, it is customary to write
We present the following new extended Riemann-Liouville-type fractional derivative operator.
Definition 3.1
The extended Riemann-Liouville fractional derivative is defined as
where ℜ(ν) < 0, ℜ(p) > 0, ℜ(m) > 0 and ℜ(μ) ≥ 0.
For n − 1 < ℜ(ν) < n, n ∈ ℕ, we write
Remark 3.2
If we take m = 0, μ = 0, and p → 0, then the above extended Riemann-Liouville fractional derivative operator reduces to the classical Riemann-Liouville fractional derivative operator.
Now, we begin our investigation by calculating the extended fractional derivatives of some elementary functions. For our purpose, we first establish two results involving the extended Riemann-Liouville fractional derivative operator.
Lemma 3.3
Let ℜ(ν) < 0, then we have
Proof
Using Definition 3.1 and 1, we have
□
Next, we apply the extended Riemann-Liouville fractional derivative to a function f(z) analytic at the origin.
Lemma 3.4
Let ℜ(ν) < 0 and suppose that a function f(z) is analytic at the origin with its Maclaurin expansion given by
Proof
Using Definition 3.1 to the function f(z) with its series expansion, we have
Since the power series converges uniformly on any closed disk centered at the origin with its radius smaller than ρ, so does the series on the line segment from 0 to a fixed z for |z| < ρ. This fact guarantees term-by-term integration as follows:
□
As a consequence we have the following result.
Theorem 3.5
Let ℜ(ν) < 0 and suppose that a function f(z) is analytic at the origin with its Maclaurin expansion given by
We present two subsequent theorems which may be useful to find certain generating function.
Theorem 3.6
For ℜ(ν) > ℜ(λ) > −
Proof
Using (30) and applying Lemmas 3.3 and 3.4, we obtain
By using (24), we can get
□
Theorem 3.7
Let ℜ(ν) > ℜ(λ) > −
Proof
Use the binomial theorem for (1 − az)−α and (1 − bz)−β. Apply Lemmas 3.3 and 3.4 to obtain
By using (25), we get
□
Theorem 3.8
Let ℜ(ν) > ℜ(λ) > −
Proof
As in the proof of Theorem 3.7, taking the binomial theorem for (1 − az)−α, (1 − bz)−β and (1 − cz)−γ and applying Lemmas 3.3 and 3.4 and taking Definition 5 into account, one can easily prove Theorem 3.8.□
Theorem 3.9
Let ℜ(ν) > ℜ(λ) > −
Proof
By using (30) and applying the Definition 2.8 for Fμ, we get
Using Theorem 3.6 for
This completes the proof.□
4 Generating functions involving the extended Gauss hypergeometric function
Here, we establish some linear and bilinear generating relations for the extended hypergeometric function Fμ by using Theorems 3.6, 3.7 and 3.9.
Theorem 4.1
Let ℜ(λ) > 0 and ℜ(β) > ℜ(α) > −
Proof
We start by recalling the elementary identity
and expand its left-hand side to obtain
Multiplying both sides of the above equality by zα−1 and applying the extended Riemann-Liouville fractional derivative operator
Uniform convergence of the involved series makes it possible to exchange the summation and the fractional operator to give
The result then follows by applying Theorem 3.6 to both sides of the last identity.□
Theorem 4.2
Let ℜ(λ) > 0, ℜ(γ) > 0 and ℜ(β) > ℜ(α) > −
Proof
Considering the following identity
and expanding its left-hand side as a power series, we get
Multiplying both sides by zα−1(1 − z)−γ and applying the definition of the extended Riemann-Liouville fractional derivative operator
The given conditions are found to allow us to exchange the order of the summation and the fractional derivative to yield
Finally the result follows by using Theorems 3.6 and 3.7.□
Theorem 4.3
Let
Proof
Replacing t by (1 − u)t in (48) and multiplying both sides of the resulting identity by uδ−1 gives
Applying the fractional derivative
The last identity can be written as follows:
Finally the use of Theorems 3.6 and 3.9 in the resulting identity is seen to give the desired result.□
5 Mellin transforms and further results
In this section, we first obtain the Mellin transform of the extended Riemann-Liouville fractional derivative operator.
Theorem 5.1
Let ℜ(ν) < 0, m > 0, ℜ(μ) ≥ 0 and
Then we have the following relation
Proof
Taking the Mellin transform and using Lemma 3.3, we have
Applying Theorem 2.6 to the last integral yields the desired result.□
Theorem 5.2
Let ℜ(ν) < 0, m > 0, ℜ(μ) ≥ 0, |z| < 1 and
Then we have the following relation
where 2F1 is a well known Gauss hypergeometric function given by (28).
Proof
Using the binomial series for (1 − z)−λ and Theorem 5.1 with λ = n yields
Then the last expression is easily seen to be equal to the desired one.□
Now we present the extended Riemann-Liouville fractional derivative of zλ in terms of the Fox H-function.
Theorem 5.3
Let ℜ(p) > 0, ℜ(μ) ≥ 0, ℜ(ν) < 0 and m > 0. Then we have
Proof
The result can be obtained by taking the inverse Mellin transform of the result in Lemma 3.3 with the aid of Theorem 2.7.□
Applying the result in Theorem 3.3 to the Maclaurin series of ez and the series expression of the Gauss hypergeometric function 2F1 gives the extended Riemann-Liouville fractional derivatives of ez and 2F1 asserted by the following theorems.
Theorem 5.4
If ℜ(ν) < 0, then we have
Theorem 5.5
If ℜ(ν) < 0, then we have
Acknowledgement
The research of J.J. Nieto has been partially supported by the Ministerio de Economía y Competitividad of Spain under grants MTM2016–75140–P, MTM2013–43014–P, Xunta de Galicia, Grants GRC2015-004 and R2016-022, and co-financed by the European Community fund FEDER.
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