ABSTRACT

In the present work, a unification of certain functions of mathematical physics is proposed and its properties are studied. The proposed function unifies Lommel function, Struve function, the Bessel-Maitland function and its generalization, Dotsenko function, generalized Mittag-Leffler function etc. The properties include absolute and uniform convergence, differential recurrence relation, integral representations in the form of Euler-Beta transform, Mellin-Barnes transform, Laplace transform and Whittaker transform. The special cases namely the generalized hypergeometric function, generalized Laguerre polynomial, Fox H-function etc. are also obtained.

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]:

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]:

The Dotsenko Function [8]:

occurs by substituting, , , , , , , , , in (1.2).

The Lommel Function defined by [9]:

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

and the Modified Struve Function [9]:

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

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,

Take

then

Thus,

Hence,

Theorem 2.2. For; and the differential recurrence relation form:

wang#title3_4:spwang#title3_4:spProof.

Consider,

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

Now consider,

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),

Now, denoting the L.H.S. of (2.4.2) by, we have

Here, introducing as a new variable of integration, by means of the relation

The further simplification gives,

as desired.

To prove (2.4.3) we begin with

Hence the result.

Now, consider

simplification of above series yields (2.4.4).

3. Mellin-Barnes Integral Representation of

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

This gives,

4. Integral Transforms of

In this section, we discussed some useful integral transforms like Euler transforms, Laplace transforms, Mellin transforms, Whittaker transformsFor the convenience, we introduce the Notation:

Theorem 4.1. (Euler(Beta) transforms)

where, and.

Proof.

Theorem 4.2. (Laplace transforms)

where, and.

Proof. We begin with

On making substitution, we get

In proving the following theorem we use the integral formula involving the Whittaker function:

Theorem 4.3. (Whittaker transforms)

where, and.

Proof. Let

then using the substitution, we get

Theorem 4.4. (Mellin transforms)

(4.4.1)

where,.

Proof. Putting in (3.1.1), we get

(4.4.2)

in which

using (1.5) and (1.6) in (4.4.2), immediately leads us to (4.4.1).

5. Generalized Hypergeometric Function Representation of

Taking, , in (1.2), we get

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 z^k in (1.2), we get

(6.1.1)

where is polynomial of degree in z^k.

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

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.

REFERENCES