Generalized extended Mittag-Leffler function and its properties pertaining to integral transforms and fractional calculus

ABSTRACT

We aim to introduce extended generalized Mittag-Leffler function (EGMLF) via the extended Beta function and obtain certain integral and differential representation of them. Further, we present some formulas of the Riemann-–Liouville fractional integration and differentiation operators. Also, we derive various integral transforms, including Euler transform, Laplace transform, Whittakar transform and K-transform. The operator and transform images are expressed in terms of the Wright generalized hypergoemetrichypergeometric type function. Interesting special cases of the main results are also considered.

KEYWORDS:

• primary 26A33
• 33C05
• 33C20
• secondary 33C65
• 33C90

1. Introduction and preliminaries

The function defined by the series representation

(1.1)

and its generalization

(1.2)

were introduced and studied by (Mittag Leffler, (1903), (Wiman, (1905), (Agarwal, (1953) and (Humbert & Agarwal, (1953), where is the set of complex numbers. The main properties of these functions are given in the book by ((Erdélyi et al., (1955), (Section 18.1) and a more extensive and detailed account on Mittag-Leffler functions is presented in ((Dzherbashyan, (1966), (Chapter 2). In particular, the functions Eqs. (1.1)(1.1) and (1.2(1.2) ) are entire function of order and type σ = 1; see, for example ((Dzherbashyan, 1966), P. 118). For a detailed account of various properties, generalizations and applications of this function, the reader may refer to an excellent work of (Dzherbashyan, (1966), (Gorenflo et al., (1988), (Kilbas & Saigo, (1995, 1996), (Attiya et al., (2023), (Bayrak & Demir, (2022), and (Zhang & Li, (2022).

(Agarwal et al., (2022). recently conducted research on the extended multi-index Mittag-Leffler functions and their relationships to integral transformations and fractional calculus. Many generalized fractional calculus operators linked to generalized and extended Mittag-Leffler functions have been studied, including those by (Gupta et al., (2016), (Kumar & Purohit, (2014), (Mishra et al., (2017), (Nisar et al., (2020), (Suthar & Amsalu, (2017), (Suthar & Purohit, (2014) and =(Suthar et al., (2020). Several researchers have examined the generalized multivariable Mittag-Leffler function and its properties, including (Jaimini et al., 2021), (Kumar, Ayant, et al., 2022; Kumar, Ram, et al., 2022), (Purohit et al., 2011). and (Suthar, Amsalu, et al., 2019; Suthar, Andualem, et al., 2019).

The series representation of Equation (1.2)(1.2) is generalized by (Prabhakar, 1971) as:

(1.3)

For it reduces to the Mittag-Leffler function given in equation Eq. (1.2)(1.2) . It is entire function of order ((Prabhakar, 1971), p.-7) and denotes the pochhammer symbol is defined as:

Further, (Salim, (2009) was introduced a new generalized Mittag-Leffler type function defined as:

In this paper, we introduced the generalized Mittag-Leffler type function in the following way:

(1.4)

which will be known as extended generalized Mittag-Leffler type function (EGMLF). Using the fact Where is the extended beta function defined by (Srivastava et al., (2012) as:

(1.5)

and is a function of an appropriately bounded sequence of arbitrary real or complex numbers defined as follows:

Some important special cases of extended generalized Mittag-Leffler type function are enumerated below:

1. If we set in Eq. (1.4)(1.4) , we obtain another form of extended generalized Mittag-Leffler type function follow as:

   (1.6)

2. On setting , and ɛ = 1, yield the known definition of (Özarslan & Yilmaz, (2014) (with c = 1)

   (1.7)

3. If we put p = 0, Eq. (1.7)(1.7) reduces to the Prabhakar’s definition (Eq. (1.3)).

4. For Eqs. (1.4)(1.4) , (1.6(1.6) ) and (1.7(1.7) ) can be expressed, respectively, in terms of the extended confluent hypergeometric functions as follows:

   (1.8)

   For our purpose, we recall the extension of Wright hypergeometric function is defined in ((P. Agarwal et al., 2018), Eq. 1.13) for as follows:

   (1.9)

2. Basic Properties of

In this section, we acquire some basic properties, including integral representation, integral and differentiation properties of the extended generalized Mittag-Leffler function.

Theorem 1.

Let such that then the EGMLF will be able to integral representation as:

(2.1)

Proof.

Using Eq. (1.5)(1.5) in Eq. (1.4)(1.4) , we obtain

Reciprocate the order of summation and integration, that is surd under the presumption given in the description of Theorem 1, we get

(2.2)

Using Eq. (1.2)(1.2) in Eq. (2.2)(2.2) , we obtain the desired result.

Corollary 2.1.

Taking in Theorem 1, we get

Corollary 2.2.

Taking in Theorem 1, we obtain

Corollary 2.3.

The following integral belongings to Eq. (1.4)(1.4) follow as:

In particular for , we have

Theorem 2.

For such that the following differentiation property for the EGMLF in Eq. (1.4)(1.4) holds true:

(2.3)

Proof.

Using the Eq. (1.4)(1.4) in right hand side of Eq. (2.3)(2.3) , we have

In view of Eq. (1.4)(1.4) , we arrived at the left hand side of Eq. (2.3)(2.3) .

In particular for , we have

Theorem 3.

Let , the EGMLF the following differentiation formula holds:

(2.4)

Proof.

Employing term by term differentiation k times on the left hand side of Eq. (2.4)(2.4) under the summation sign, we get

In particular for , we have

3. Integral transform of

For the convenience of the reader, we provide here the basic definitions and related notations which is necessary for the understanding of this study.

Definition 3.1.

Euler Transform ((Sneddon, 1979))

The Euler transform of a function f(z) was defined as:

(3.1)

Definition 3.2.

Laplace Transform ((Sneddon, 1979))

The Laplace transform of a function denoted by was defined by the equation

(3.2)

provided the integral (3.2(3.2) ) is convergent and that the function is continuous for t > 0 and of exponential order as (3.2(3.2) ) may be symbolically written as:

Definition 3.3.

Whittakar Transform ((Whittaker & Watson, 1962))

(3.3)

where and is the Whittakar confluent hypergeometric function

where is defined by

Definition 3.4.

K-Transform ((Erdélyi et al., 1954))

This transform was defined by the following integral equation

where is the Bessel function of the second kind defined by ((Choi et al., (2016), (p. 332)

where is the Whittakar function defined in Equation (3.3)(3.3) .

The following result given in ((Mathai et al., 2010), p. 54, Eq. 2.37) will be used in evaluating the integrals

(3.4)

Further, we evaluate the following Euler transform, Laplace transform, Whittakar transform and K-transform of EGMLF.

Theorem 4.

(Euler Transform)

Let be such that

(3.5)

where    .

Proof.

Using (3.1(3.1) ) and (1.4(1.4) ), it gives

In accordance with the definition of (1.9(1.9) ), we obatain the desired result (3.5(3.5) ).

Corollary 3.1.

When , Eq. (3.5)(3.5) reduces to another form of extended generalized Mittag-Leffler type function in the following form

Corollary 3.2.

For and ɛ = 1, Eq. (3.5)(3.5) reduces in the following form

Corollary 3.3.

If we set , Eq. (3.5)(3.5) reduces to extended confluent hypergeometric functions in the following form

Theorem 5.

(Laplace Transform)

Let and be such that

(3.6)

where   .

Proof.

Using (3.2(3.2) ) and (1.4(1.4) ), it gives

In accordance with the definition of (1.9(1.9) ), we obatain the desired result (3.6(3.6) ).

Corollary 3.4.

When , Eq. (3.6)(3.6) reduces to another form of extended generalized Mittag-Leffler type function in the following form

Corollary 3.5.

For and ɛ = 1, Eq. (3.6)(3.6) reduces in the following form

Corollary 3.6.

If we set , Eq. (3.6)(3.6) reduces to extended confluent hypergeometric functions in the following form

Theorem 6.

(Whittakar Transform)

Let ; be such that

(3.7)

where    .

Proof.

Using (3.3(3.3) ) and (1.4(1.4) ), it gives

In accordance with the definition of (1.9(1.9) ), we obtain the desired result (3.7(3.7) ).

Corollary 3.7.

When , Eq. (3.7)(3.7) reduces to another form of extended generalized Mittag-Leffler type function in the following form

Corollary 3.8.

For and ɛ = 1, Eq. (3.7)(3.7) reduces in the following form

Corollary 3.9.

If we put , Eq. (3.7)(3.7) reduces to extended confluent hypergeometric functions in the following form

Theorem 7.

(K-Transform)

Let ; be such that

(3.8)

where    .

Proof.

Using (3.4(3.4) ) and (1.4(1.4) ), it gives

In accordance with the definition of (1.9(1.9) ), we obtain the result (3.8(3.8) ).This completes the proof of the theorem.

Corollary 3.10.

When , Eq. (3.8)(3.8) reduces to another form of extended generalized Mittag-Leffler type function in the following form

Corollary 3.11.

For and ɛ = 1, Eq. (3.8)(3.8) reduces in the following form

Corollary 3.12.

If we set , Eq. (3.8)(3.8) reduces to extended confluent hypergeometric functions in the following form

4. Fractional calculus approach of

In this section, we derive a little interesting properties of EGMLF associated with the operators of Riemann–-Liouville fractional integrals and derivatives are defined for  x > 0 (See, for details (Kilbas et al., 2006)):

and

respectively, where is the integral part of , The following lemma is needed in the sequel ((Salim, 2009), Eq. 2.44).

Lemma. Let and then

1. If then

   (4.1)

2. If then

   (4.2)

where such that

Theorem 8.

Let be such that    and the following integral formula holds true:

(4.3)

Proof.

Let us denote the left-hand side of (4.3(4.3) ) by I₁. Using Eq. (1.4)(1.4) , we have

(4.4)

Interchanging the integration and the summation in Eq. (4.4)(4.4) , we get

Applying the relation (4.1(4.1) ) in Lemma, we obtain

In view of Eq. (1.9)(1.9) , we arrived at the desired result.

Theorem 9.

Let such that , ,   and the following integral formula holds true:

(4.5)

Proof.

Denoting the left-hand side of (4.5(4.5) ) by I₂. Using Eq. (1.4)(1.4) , we have

(4.6)

Interchanging the integration and the summation in Eq. (4.6)(4.6) , we get

Applying the relation (4.2(4.2) ) in Lemma, we obtain

In view of Eq. (1.9)(1.9) , we arrived at the desired result.

Theorem 10.

Let be such that    and then the following fractional differentiation formula holds true:

(4.7)

Proof.

Let I₃ denote the left-hand side of (4.7(4.7) ). Using Eq. (1.4)(1.4) , we have

Applying the relation (4.1(4.1) ) in Lemma, we obtain

By interchanging the differentiation and summation, we get

In view of Eq. (1.9)(1.9) , we arrived at the desired result.

Theorem 11.

Let such that ,  , , then the fractional differentiation formula of EGMLF is given by

(4.8)

Proof.

Denoting the left-hand side of (4.8(4.8) ) by I₄. Applying Eq. (1.4)(1.4) , we have

Applying the relation (4.1(4.1) ) in Lemma, we obtain

By interchanging the differentiation and summation, we get

In view of Eq. (1.9)(1.9) , we arrived at the desired result.

4.1. Special cases

In this section, we derive in Corollaries 4.1 - 4.12 with some specialized to yield the corresponding formulas by using known extended generalized Mittag-Leffler type function, generalized Mittag-Leffler type function and extended confluent hypergeometric functions.

(i) If we employ the same method as in proofs of Theorems 8 - 11, we obtain the following four corollaries with the help of (1.6(1.6) ) which is known another form of extended generalized Mittag-Leffler type function. For the conditions of , the above Theorems reduce to:

Corollary 4.1.

Let be satisfied, the following integral formula holds:

Corollary 4.2.

Let be satisfied, the following integral formula holds:

Corollary 4.3.

Let be satisfied, the following differentiation formula holds:

Corollary 4.4.

Let be satisfied, the following differentiation formula holds:

(ii) If we put and ɛ = 1, the similar way as in proofs of Theorem 8 - 11, we obtain the following four corollaries with the help of (1.7(1.7) ) below as:

Corollary 4.5.

Let the condition of Theorem 8 be satisfied, the following integral formula holds:

Corollary 4.6.

Let the condition of Theorem 9 be satisfied, the following integral formula holds:

Corollary 4.7.

Let the condition of Theorem 10 be satisfied, the following differentiation formula holds:

Corollary 4.8.

Let the condition of Theorem 11 be satisfied, the following differentiation formula holds:

(iii) Similarly, putting p = 0 in Corollaries 4.5 - 4.8, we get the fractional integral and differentiation formulas involving the Prabhakar-type (Prabhakar, 1971) Mittag-Leffler function. We omit the details.

(iv) If we set in proofs of Theorem 8 - 11, we obtain the following four corollaries with the help of (1.8(1.8) ) which is known extended confluent hypergeometric functions as follows:

Corollary 4.9.

Let the condition of Theorem 8 be satisfied, Eq. (4.3)(4.3) reduces in the following form:

Corollary 4.10.

Let the condition of Theorem 9 be satisfied, Eq. (4.5)(4.5) reduces in the following form:

Corollary 4.11.

Let the condition of Theorem 10 be satisfied, Eq. (4.7)(4.7) reduces in the following form:

Corollary 4.12.

Let the condition of Theorem 11 be satisfied, Eq. (4.8)(4.8) reduces in the following form:

5. Concluding remark and discussion

The properties, integral transform and images of fractional calculus of the newly defined extended generalized Mittag-Leffler type function are investigated here. Various special cases of the derived results in the paper can be evaluate by taking suitable values of parameters involved. For example, if we set ɛ = 1 in (1.4(1.4) ), we obtain the some result due to (Parmar, 2015).

Authors contributions

All authors contributed equally to the present investigation. All authors read and approved the final manuscript.

Acknowledgements

The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Notes on contributors

A. Padma

Dr. A. Padma is an assistant Professor, Dept. of Mathematics, Chaitanya Bharathi Institute of Technology, Gandipet, Hyderabad, Telangana. My research interests include Number Theory, Fixed Point Theory, Real Analysis. I had published 16 research article in international and national level.

M. Ganeshwara Rao

Dr. M. Ganeshwar Rao is Professor, Dept. of Mathematics, Chaitanya Bharathi Institute of Technology, Gandipet, Hyderabad, Telangana. My research interests include Number Theory and Real Analysis.

Biniyam Shimelis

Biniyam Shimelis is an Assistant Professor, Department of Mathematics, College of Natural Science, at Wollo University, Dessie, Amhara Region, Ethiopia. I have acquired a M.Sc. in Analysis with 15 years of teaching experience and 8 years of research experience. My research interests include Special functions, Fractional Calculus, Integral transforms, Basic Hypergeometric Series, Geometric Function Theory and Mathematical Physics. I had also organized a diverse range of workshops/conferences in the fields of mathematical sciences, engineering innovations and a number of seminars on mathematical discipline in order to enrich their knowledge and the skills of the student community.