On a Unification of Generalized Mittag-Leffler Function and Family of Bessel Functions ()
1. Introduction
In the present work, we propose an extension of a generalization of the Mittag-Leffler function due to A. K. Shukla and J. C. Prajapati [1], defined as
(1.1)
where
;
and
. This is an entire function of order
if
and absolutely convergent in
if
. In fact (1.1) contains the
-Mittag-Leffler function [2],
- the generalized Mittag-Leffler function [3] and the function
due to Prabhakar [4].
Gorenflo et al. [5], Saigo and Kilbas [6] studied several interesting properties of these functions.
Another generalization of Mittag-leffler function due to T. O. Salim [7], given by
(1.1’)
where
and
.
We state below the extended version in the form:
(1.2)
where
,
,
. The function defined by (1.2) reduces to the one in (1.1) and (1.1’) if
,
,
,
and
,
,
,
,
respectively.
It is noteworthy that the function in (1.2), besides containing the generalizations of the Mittag-Leffler function, also includes certain functions belonging to the family of Bessel function. To see this, take
,
,
,
,
,
, and replaced z by
in (1.2), then we find the well known Bessel function [8]:
![](https://www.scirp.org/html/17-5300300\3622d383-7317-413e-bdfa-ac0a5786997e.jpg)
When
,
,
,
, and z is replaced by
then we get the Bessel Maitland Function [8] given by
For
,
,
,
,
,
,
,
,
,
, and z is replaced by
, we obtain the Generalized Bessel Maitland function [8]:
![](https://www.scirp.org/html/17-5300300\b72bc103-d256-4235-b5f0-ca55fea81ee4.jpg)
The Dotsenko Function [8]:
![](https://www.scirp.org/html/17-5300300\a7c0e5af-e3dc-42fd-bba4-4963314d8b34.jpg)
occurs by substituting
,
,
,
,
,
,
,
,
,
in (1.2).
The Lommel Function defined by [9]:
![](https://www.scirp.org/html/17-5300300\d0f5b822-8337-450b-9751-e22ef902626e.jpg)
is the special case
,
,
,
,
,
,
,
,
,
, and z is replaced by
of (1.2). On making substitutions
,
,
,
,
,
,
,
,
,
, and
in (1.2), provides us respectively, the Struve Function
[9] given by
![](https://www.scirp.org/html/17-5300300\46395f86-939b-4ea5-94e2-db028e954dd8.jpg)
and the Modified Struve Function [9]:
![](https://www.scirp.org/html/17-5300300\9f5f0403-18ae-4f13-ad0f-5051062b52b0.jpg)
In what follows, we shall use the following definitions and formulas. Euler (Beta) transform [10]:
(1.3)
Laplace transform [10]:
(1.4)
Mellin-Barnes transform [10]:
(1.5)
then
(1.6)
Incomplete Gamma function [11]:
(1.7)
The generalized hypergeometric function is denoted and defined by [11]
(1.8)
where
are neither zero nor negative integers, and
.
The series is convergent for 1)
if
2)
if ![](https://www.scirp.org/html/17-5300300\4234cc85-e0b4-4d9b-b16c-bd685fa5407e.jpg)
Wright generalized hypergeometric function [12]:
(1.9)
Laguerre polynomial [12]:
(1.10)
2. Main Results
In this section, we prove the following results for the function defined in (1.2).
Theorem 2.1. The series represented by the function
converges absolutely for
.
Proof: Consider,
![](https://www.scirp.org/html/17-5300300\b9cce0b9-f607-4472-a23a-7b63505af741.jpg)
Take
![](https://www.scirp.org/html/17-5300300\9a38522c-7236-4fca-87b5-c660110d0b27.jpg)
then
![](https://www.scirp.org/html/17-5300300\1f70798c-78ac-45bd-a627-5859eb5981fa.jpg)
Thus,
![](https://www.scirp.org/html/17-5300300\eb77e24d-289d-43ae-8818-5809f4322b50.jpg)
Hence,
![](https://www.scirp.org/html/17-5300300\0996456e-26b5-4f69-bfe3-537cf7c11afc.jpg)
Theorem 2.2. For
;
and
the differential recurrence relation form:
![](https://www.scirp.org/html/17-5300300\5d0fcef5-3663-41c7-aca2-002582dd22c8.jpg)
wang#title3_4:spwang#title3_4:spProof.
Consider,
![](https://www.scirp.org/html/17-5300300\583f5da2-c968-46b0-ad34-abde889e5cad.jpg)
As the series given in (1.2) converges uniformly in any compact subset of
, the use of term by term differentiation under the sign of summation leads us to the following theorem.
Theorem 2.3. If
,
,
and
then
(2.3.1)
(2.3.2)
Proof. Consider
![](https://www.scirp.org/html/17-5300300\22d3a92b-2a71-4d22-be91-1f0686256553.jpg)
Now consider,
![](https://www.scirp.org/html/17-5300300\0f4d42e1-d0a9-4203-b217-ad78f86f3c1c.jpg)
Next, taking
in the Euler (Beta) transform (1.3), one finds the following Theorem 2.4. If
,
and
then
(2.4.1)
(2.4.2)
(2.4.3)
(2.4.4)
wang#title3_4:spwang#title3_4:spProof.
In (2.4.1),
![](https://www.scirp.org/html/17-5300300\287d4410-cb10-45cc-95f5-91b0c3ef0311.jpg)
Now, denoting the L.H.S. of (2.4.2) by
, we have
![](https://www.scirp.org/html/17-5300300\5c4bffbc-973d-44f5-91a5-a4f6b69db14a.jpg)
Here, introducing
as a new variable of integration, by means of the relation
![](https://www.scirp.org/html/17-5300300\516712ee-f286-4f8e-911d-8278f20abbaf.jpg)
The further simplification gives,
![](https://www.scirp.org/html/17-5300300\3933bcb5-0504-4d97-b927-970d18355663.jpg)
as desired.
To prove (2.4.3) we begin with
![](https://www.scirp.org/html/17-5300300\9f12350d-f46b-420f-a08b-857264cf679b.jpg)
Hence the result.
Now, consider
![](https://www.scirp.org/html/17-5300300\ae467131-27de-408d-8c24-6e66b36dc998.jpg)
simplification of above series yields (2.4.4).
3. Mellin-Barnes Integral Representation of ![](https://www.scirp.org/html/17-5300300\f0c7d9c0-f1bc-4e82-ba72-694a8f9e66dd.jpg)
Theorem 3.1. Let
;
,
and
,
. Then the function
is represented by the Mellin-Barnes integral as
(3.1.1)
where
the contour of integration beginning at
and ending at
, and indented to separate the poles of integrand at
for all
(to the left) from those at
for all
(to the right).
wang#title3_4:spwang#title3_4:spProof.
We shall evaluate the integral on the R.H.S. of (3.1.1) as the sum of the residues at the poles
In fact, in view of the definition of residue, we have
![](https://www.scirp.org/html/17-5300300\5cb88c3e-522c-4c32-ab53-3f1080d473d4.jpg)
This gives,
![](https://www.scirp.org/html/17-5300300\d75000ad-42c4-4e52-8029-e42851bfaf9d.jpg)
4. Integral Transforms of ![](https://www.scirp.org/html/17-5300300\020b63ff-dab5-488b-b6e6-bd3c77883814.jpg)
In this section, we discussed some useful integral transforms like Euler transforms, Laplace transforms, Mellin transforms, Whittaker transformsFor the convenience, we introduce the Notation:
![](https://www.scirp.org/html/17-5300300\433de119-f2ac-40de-a779-b88dd9fd2616.jpg)
Theorem 4.1. (Euler(Beta) transforms)
![](https://www.scirp.org/html/17-5300300\1db4ac9e-6cbb-43c2-80c0-725c17d1855b.jpg)
where
,
and
.
Proof.
![](https://www.scirp.org/html/17-5300300\2472cc8d-7eac-4a8c-bd20-4e0af7772da9.jpg)
Theorem 4.2. (Laplace transforms)
![](https://www.scirp.org/html/17-5300300\addb3985-aa1d-4c90-b528-5253d8fdefd9.jpg)
where
,
and
.
Proof. We begin with
![](https://www.scirp.org/html/17-5300300\ae1a9c0c-5518-498f-89bd-cb408c526602.jpg)
On making substitution
, we get
![](https://www.scirp.org/html/17-5300300\162a9b66-13f5-4c93-9b9c-c7795d230753.jpg)
In proving the following theorem we use the integral formula involving the Whittaker function:
![](https://www.scirp.org/html/17-5300300\2009aa10-1f6d-48f9-b8de-a553d72092ce.jpg)
Theorem 4.3. (Whittaker transforms)
![](https://www.scirp.org/html/17-5300300\708ac455-2447-48e5-ad7e-89d342059add.jpg)
where
,
and
.
Proof. Let
![](https://www.scirp.org/html/17-5300300\4ebc5caa-71bb-4e46-8fd1-83e36bb39377.jpg)
then using the substitution
, we get
![](https://www.scirp.org/html/17-5300300\5de583d5-941c-4743-8212-c9a6da9b1621.jpg)
Theorem 4.4. (Mellin transforms)
(4.4.1)
where
,![](https://www.scirp.org/html/17-5300300\943a07f4-43f8-4f7c-817f-3cadf051e6ff.jpg)
.
Proof. Putting
in (3.1.1), we get
(4.4.2)
in which
![](https://www.scirp.org/html/17-5300300\8cde34c4-696e-4335-b5c9-4eedff90e504.jpg)
using (1.5) and (1.6) in (4.4.2), immediately leads us to (4.4.1).
5. Generalized Hypergeometric Function Representation of ![](https://www.scirp.org/html/17-5300300\de66241a-ed4b-458b-aa09-f7547dbec2e8.jpg)
Taking
,
,
in (1.2), we get
![](https://www.scirp.org/html/17-5300300\f0e97f3e-bbcc-4717-9a0e-881e6f9dd175.jpg)
where
is a n-tuple
.
6. Relationship with Some Known Special Functions (Generalized Laguerre Polynomial, Fox H-Function, Wright Hypergeometric Function)
6.1. Relationship with Generalized Laguerre Polynomials
Putting
,
,
,
,
,
,
and replacing
by
and z by zk in (1.2), we get
(6.1.1)
where
is polynomial of degree
in zk.
In particular,
so that
(6.1.2)
6.2. Relationship with Fox H-Function
From (3.1.1), we have
(6.2.1)
6.3. Relationship with Wright Function
If
,
,
(1.2) can be written as
(6.3.1)
from (1.9) for (6.3.1), we get
![](https://www.scirp.org/html/17-5300300\41a3dc6f-a218-409d-ae6f-7da5ec24a8ff.jpg)
7. Summary
In Section 1, an extended version of Mittag-Leffler function of 10 indices established as an Equation (1.2) including with some necessary information of Bessel function, some well-known integral transforms and generalized hypergeometric functions with their family. Results obtained in Sections 2 to 6 are interesting generalizations of (Shukla and Prajapati [1]) and stimulate the scope of further research in the field of generalization MittagLeffler function.
8. Acknowledgements
This paper dedicated to our beloved great Mathematician Gösta Mittag-Leffler. The authors would like to thank the reviewers for their valuable suggestions to improve the quality of paper.